Thanks again, Ray. Oddly enough, I think the problem may have resolved itself! I left my PC running for some hours, and on return I found it seemed to have restarted itself, but had been interrupted by Spybot-SD Resident, which reported that a registry entry for Shockwave Updater was being deleted. I decided to let it do so, and there was an immediate request for it to be replaced by what I assume to be a new version. This I declined. Next time I started the machine, the unwanted icon had vanished. From my Internet searches in other forums, it seems plenty of others have had the same thing happen, so maybe this is the way to fix it ....
I've downloaded the trial version of Speed Startup, and will try it out sometime.