When I close my PC I am increasingly getting a message that Ad-Aware Tray ....... is not responding and I have to "end task" so as to shut down.
I am using Ad-Aware Plus Anniversary Edition.

Will it be as simple as reinstalling it ?

If you try a reinstall, do a clean uninstall first to clean out all files and registry entries. Or you could right-click the tray icon and exit before shutting down, albeit that's a hassle. FYI, as a paid Plus user, you can contact Lavasoft Support directly via the link in my signature - they'd know better the cause and solution.