Thank you, Henry Crun and mixture. I have explained myself inadequately, I'm sorry.
I've tried a total of five different mice, the original wireless USB one and four replacements (one different wireless USB of a different make, two distinct cable USB mice and then one cable PS2 mouse. In fact each of the five mice is from a different manufacturer).
The problem wasn't there at all a couple of months ago and only occurs in the two situations that I mentioned - in every other operation all of the five mice work perfectly. I honestly cannot understand that the same problem is at all likely to have appeared simultaneously in the guts of all five very different mice at precisely the same time.
I was expecting a diagnosis of some shared circuitry or card in my computer - shared by all five mice, I mean - or perhaps the click speed or some other user setting needs to be changed.
Perhaps I'm barking up the wrong tree but I thought that I'd tried enough actual mice, all different and pretty much covering the whole range of what is available (while adjusting the hardware setting in CP for each in turn), for the problem to lie within all of the mice. It's not the cost of a mouse which is swaying me - indeed one of the cable USB mice is a brand new one.
Since the problem occurs only on one tiny part of the BBC news site and not on apparently identical other bits I thought at first that it was down to some update or change by the BBC but the thumbnail problem turning up identically and at the same time on lots of different sites seems to put the lie to that.