Thanks, Asyn
That certainly looks like a plausible explanation for the sudden rise in popups of this general kind. I presume that these will gradually become less common as server certificates are renewed.
The specific issue with browser-update.avast.com remains, however, and it is troubling as I have had seven instances in the last half-hour, so I'm afraid I need to return to the questions in my original posting, viz.:
Is this a problem that I can overcome somehow? If so, could someone please advise?