@DavidR: Thanks for your help. I think I've got things under control now. Yes, I had installed the Sony Puppy Suite a long time ago, as I'd mentioned in my original post. I've now switched AutoSandbox mode to Ask from Auto, as you suggest. I also added PsTrayIcon.exe to the (short) list of files that will be excluded from automatic sandboxing. After rebooting, I no longer get the Avast pop-up.
By the way, when you say "it isn't a false positive as such", I'm sure you're technically correct. All I know, as an unsophisticated Avast user, is that I've had both PsTrayIcon.exe and Avast AV running on my system for years without Avast complaining about PsTrayIcon.exe. It was only in the last few weeks that Avast started displaying the unusual popup about PsTrayIcon.exe. I suppose maybe that's a recent "improvement" in Avast.
Well, I'm OK with this "improvement", now that I understand what to do about it. Thanks again!