Hi again everyone; sorry about the delay and the sloppy haphazard way I am conducting this thread.
I promised feedback on the docked external HDD installation. That went as one would expect; no abnormalities noticed. As on the laptop I first installed Manjaro and then proceeded to install Solus Plasma next to it allowing both OS's roughly half of the disk space (250GB each). I then proceeded to re-install Solus from the same live ISO USB, but utilizing the entire disk this time... no problems. Then I re-installed Manjaro as before, but now replacing Solus, and again: no problem.
Back to the laptop, here is a summary of what happens trying to start up:
1.Power Button by itself gives the hp splash screen with a notification at the bottom:
"Press the ESC key for Startup Menu"

  1. Power Button > ESC gives the hp splash screen with a notification at the bottom:
    "ESC ... Pause Startup"

  2. Power Button > F1 gives the hp splash screen with a notification at the bottom:
    "F1 ... System Information"

  3. Power Button > F2 gives the hp splash screen with a notification at the bottom:
    "F2 ... Startup Check"

  4. Power Button > F3 through F8 gives the hp splash screen with a notification at the bottom:
    "Press the ESC key for Startup Menu"

  5. Power Button > F9 gives the hp splash screen with a notification at the bottom:
    "F9 ... Change Boot Device Order"

  6. Power Button > F10 gives the hp splash screen with a notification at the bottom:
    "F10 ... BIOS Setup Options"

  7. Power Button > F11 gives the hp splash screen with a notification at the bottom:
    "F11 ... System Recovery"

  8. Power Button > F12 gives the hp splash screen with a notification at the bottom:
    "Press the ESC key for Startup Menu"

None of the above result in any further action, choices or possibilities... all of them end and get stuck at that stage.
Pressing any key or sequence of keys after this a number of times ( I did not count) simply results in a continuous "Beeeeep".

All this seems to suggest that the actual BIOS is not 'broken', but may perhaps only need resetting or something like that??

I really hope that this additional bit helps.

Many thanks for your efforts to come up with a solution this far!

F11 all by itself gives me a boot window to choose bios or my two (or sometimes one) OS's. I read you 4 times and unsure if we are on laptop hdd now and not external hdd? is external unplugged on all this? what OS is on the laptop. It's either manjaro or solus.
bootloader get mangled?
are you wiping the disk after each install (gparted) and reformatting GPT?
you can get the basic HP splash but everything is dead after that?
CTRL+ALT+F2 (or 3 or 1 etc) upon booting should give you a terminal window. If you can get that you are on the road to wellness.
2 cents
edit @janetlox

My summary of what happens when I try to start the laptop is the laptop unplugged with no externals like USB sticks, external HDD's or anything attached, and no operating system installed. Manjaro and Solus were on the laptop hard disk when I tried to replace both of them with Solus ... and got stuck halfway through with the notification that the install had failed, the disk had been 'wiped', and to restart to try again. I did try to restart which is the stage I am now at and do not know how to proceed from ... as described, the hp logo and splash screen with bottom notification are displayed every time, without any option to proceed further.
The docked external hard drive experience is on my Dell Windows 10 main PC with a USB docking station plugged in to run old hard disk drives for making backups and trying various OS's like Manjaro and Solus, and yes, I even had both of those on that hard disk to try and duplicate the failed efforts of the HP laptop.
At no stage have I wiped or partitioned the laptop hard disk. Being a novice usually leave those things for the installer to take care of. Likewise reformatting the GPT is a not to be trodden on minefield for me ... I wouldn't know where to start or even what I was trying to accomplish.
I hope this helps?

@brent
I failed to mention that I tried your final suggestion: "CTRL+ALT+F2 (or 3 or 1 etc) upon booting should give you a terminal window.", but sadly no terminal window -- or anything else for that matter.
Indeed, whatever I try, I get the basic HP splash screen with everything 'dead' after that.

    janetlox

    Have you tried to pull the hard drive and boot the machine without it? It shouldn't cause your issue but you never know. The other option could be a memory chip issue.

    I don't know if you have removed a hard drive or not before, but that video above shows the process at about 50 seconds in.

      I have never removed a hard drive or anything else from a laptop before, and would rather leave that as a last resort.
      I do remember reading somewhere that disconnecting and replacing the CMOS battery (if that is what it is called) will definitely reset the BIOS to its original (factory) settings. Should that maybe be my next move? If so, I might as well do the hard drive removal trick then?

        janetlox

        Yes, removing the CMOS battery may work just as well, as it drains the bios memory , resetting everything back to defaults. Normally they are buried in a corner of a motheroard or soldered on, but in your case it looks as though it is under the long centre panel on the bottom of the machine. It looks like it's a cr232 battery, just at the base of the memory chips.

        Pull it out for 10 minutes or so along with the main battery, and then see what it does.

          I removed the CMOS battery, and replaced it about an hour later.
          Upon starting the laptop again there was a notification of which I do not remember the exact wording, but it boils down to the fact that the BIOS or BIOS settings had indeed been changed, that I could check F10 BIOS options to see details and that the system would restart ... which it did before I could even finish reading it all ...
          After that restart the laptop was back exactly as it had been before -- it would display the logo splash screen with notifications exactly as before and likewise fail to proceed any further.

          I think I will have to take a good look at more technical (BIOS- and PC Startup) fora and/or specific hp write ups, and worry about Solus again as soon as I have found a solution, not that Solus is likely to present any problems :-)

          A BIG THANK YOU to everyone for the help in trying to solve this problem -- it seems to be a tricky one.

          Regards, janetlox

          The easiest way to boot your laptop from a usb is plug in your boot media and mash away at the F9 key directly after switching on, then select your boot media from the next menu. If there are no available drives to choose from this may be a RAM failure but it could just be a misconfiguration between UEFI and Legacy boot.
          First. What software are you using to build your bootable USB? (Ventoy is my favourite, but Rufus and Etcher are also very good.)
          Second. Can you dig about in the BIOS a bit and determine if your machine has been set to Legacy or UEFI.
          Sorry these are very necessary steps. This link might help.
          Some of the symptoms you are describing sound a lot like you are trying to boot a machine that is set to legacy boot with a stick set up for UEFI although it is also possible that everything is set up right and your USB stick has corrupted and needs to be rewritten or replaced.

          A final thought that could simplify this might be to burn the ISO to an old fashioned DVDR using the "Burn ISO to Disk" option from almost any disk authoring software. Then try F9 at switch on as before and look for your DVD drive to boot from. Please choose a lowish speed to write at for maximum repeatability.

          You can prove a whole lot by removing the hard drive completely and booting into the live media. Despite your misgivings about this, it is really worth a try.

          Thanks BuzzPCSOS, I am using a 64GB Ventoy stick, but have tried ImgBurn prepared sticks as well with no luck.
          The big issue is that I can start up using F9 through F12 and will unfailingly get the HP splash screen, but cannot proceed any further beyond that, so trying to dig around in the BIOS is exactly what I am trying to do.
          I have also tried the DVD option with Solus ISO, but to select the optical drive have to go through the same steps as mentioned above... which does not work.

          The only option left is as you suggest is removing the hard drive completely etc., but you are right about my MISGIVINGS about that. Anyhow I ill have to do something, so I'll have a good look at that video again and hope to gather enough guts to enter the wilderness of the laptop innards...

          One question though: will removing the hard drive serve any purpose in this case since there is supposedly nothing on there (wiped completely by the Solus installer)?

          I removed the hard drive and powered up without it... Voila, a whole lot of options and a message that no OS could be found and to please install one on the hard disk... which is going to be hard to do with the hard disk removed and the only way to access any options IS with the HDD removed.
          Any suggestions as to how I should proceed from here? I am sorry for all the hassles, but I am very pleased that at last I have gotten to a stage other than a dead end.
          I am hesitant to re-install the hard drive now to see what happens and would rather await further advice.

            janetlox

            Hi again, It sounds like you have a corrupt install on your hard drive which is preventing the machine from booting properly. So I would suggest that you try to reinstall windows on the machine to start with, as you are familiar with that operating system. If you search on google for windows ISO you should be able to find the area of the microsoft site that will let you create a windows install usb on your other machine.

            On your HP Pavilion press F10 to get into the bios and go to the "BOOT" section There you should be able to change the order that the laptop checks for boot devices. Move the USB / Removable Disk, I'm not sure what it will be listed as, option to the top of the list so that the usb is checked before any other device. Then replace the hard disk and try to boot from the windows USB. Hopefully you should be able to get your system back up and running, and then try again with solus.

            I had a quick look online, but couldn't find a detailed explanation of your BIOS, but I did find this quick sort of guide.

            https://www.4winkey.com/windows-tips/how-to-enter-access-get-into-bios-on-hp.html

              If you access the BIOS (F10) you have an option to adjust the boot order so that your DVD or USB is the first boot device on the list. That way you will not need to access the F9 menu to boot the Solus live session. Then use disks application to fully format the drive. Remember to set the boot order back to hard drive first once the install has completed.
              I recall a mention of how your laptop had been a little sluggish, it is likely that the errors you have been experiencing during install are due to a hard drive at the end of its service life. Now you have taken the jump of removing the hard drive it might be a good idea to replace the whole thing. An SSD would make a good substitute and would improve drive access time by about 6x which is really noticeable. A 120Gb SSD should be available for about £15 (GB pounds).
              Good Luck.

              With the hard drive removed I can access external drives, change the boot order etc., but whatever the boot sequence, as soon as the hard disk is replaced, I can once again not access external drives and keep getting the dead end hp splash screen.
              My conclusion is that the hard drive has reached the end of its life, so I opted to order a 2.5" sata SSD which should be here tomorrow, and I am hopeful that that will finally solve this issue.
              Installing a version of Windows 10 will not serve any purpose since I am not at all familiar with MS/Windows systems even though I have been running Windows as my daily machine for as long as I can remember. What I do remember is that whenever there was a problem, I mostly only arrived at a solution after rummaging through my Linux stuff.
              I only installed Windows 10 once and recall that the install- and post install exercises drove me mad!
              In the end Linux may not appear as 'polished' as Windows, but for 'novices' like me the obvious choice is for the system that just works and a community one can approach for help and advice.
              It may take a while, but the result of this exercise will eventually appear here :-)

              I installed the new SSD, installed Solus and performed the first compulsary update plus a few additional applications, ... and I am writing this using Solus KDE on a sweet running hp laptop! I could hardly be happier.
              A big, hearty and warm THANK YOU to all of you for your efforts helping me achieve this happy end. it is much appreciated!!!
              Kind regards,
              Jan (janetlox)