On reboot winpatrol reported the mystery program asking to be added to the start menu.
Here are the screen shots of winpatrol, in case someone can spot any difference;
edit: Disclaimer; BTW, the laptop and its OS and content are 8 or 10 years old but I've only had it for a few months.

I'm thinking this might explain what you're experiencing: https://forums.adobe.com/thread/1654402

[...]Speed Launcher is adding a HKCU RunOnce entry over and over again and assigning a random number to it.  This is then of course alerting WinPatrol who is monitoring new startup programs[...]

I'm thinking this might explain what you're experiencing: https://forums.adobe.com/thread/1654402

Quote

[...]Speed Launcher is adding a HKCU RunOnce entry over and over again and assigning a random number to it.  This is then of course alerting WinPatrol who is monitoring new startup programs[...]

The only Adobe entry I could find in winpatrol's Start list was AdobeARM, which I have thankfully disabled. It is reported to be unnecessary to normal operation of the OS, can be started when needed anyways, and -yes- is reported to be variously problematic. This may indeed be the culprit; and as a legitimate program, no anti-virus or anti-pup scan would find it. I can't thank you and the other respondents enough for all your help! 

Quote from: Dirhael on February 15, 2016, 06:06 PM

I'm thinking this might explain what you're experiencing: https://forums.adobe.com/thread/1654402

Very good catch!

Quote from: holt on February 16, 2016, 05:27 AM

The only Adobe entry I could find in winpatrol's Start list was AdobeARM...

"Adobe Speed Launcher" shows as the 3rd item in your "winpatrol 7.jpg" screen capture. Maybe it doesn't last long in the list due to whatever bug it has.  The directions in Dirhael's link should fix your problem.

And look at the command- I didn't pay enough attention.  It appears to be a number.  WinPatrol strikes again!

Every time I click on any .pdf file, WinPatrol immediately pops up with a random number string that is a mystery program it says is asking to be added to the Start menu; it's definitely Adobe-related. So I just click on 'reject' and it goes away and quits bugging me; but you have to disable the above-mentioned Adobe software -AdobeARM or Adobe Speed Launcher- from the Start menu to prevent it bugging you on a daily basis. Once again, many thanks to everyone who helped me so patiently, and especially to Dirhael.

@holt, did you install the recommended patch?
(https://helpx.adobe.com/acrobat/kb/runonce-key-created-every-user.html via Dirhael's link above)

Somehow I missed that. Got it. Thanks tomos!