My suggestion that the Apple technician solved the "aggravating pop-up" issue was not quite correct. By getting iCloud to acknowledge my Apple ID, he only solved the problem of it popping up continuously - simply downgrading the problem from a constant one to a periodic one similar to Tacit's.
I've tried signing out of iCloud (letting it delete all the synced info from my Mac) and signing back in, which solves the problem for a day or two, but like rainclouds on a day off it just keeps coming back.
Three times since my last posting I've got the following dialogue box when first booting my Mac.
This Mac can’t connect to iCloud because of a problem with "email address".
Open iCloud preferences to fix this problem.When I open the iCloud preferences, I get this dialgue box.
Enter the Apple ID password for "email address"
Enter your password to continue using iCloud on this Mac.I enter the Apple ID Password, it signs in, I close the dialogue box and carry on.