Thanks for your prompt reply, but as I said, (or perhaps should have made clearer) I CAN sign on with a current ID. The problem is that when I start up my desktop I have to go through a routine where I am asked to verify my Apple ID with a no longer existing email and password, and the same for iCloud etc. When I try to sign in with this defunct account name I am told that it does not exist in the system (of course). Pressing "Not Now" only results in the window popping up again at the same point. The problem is even worse on the iPhone which keeps chucking back the same window many, many times (sometimes 10 or 12 or more. Very inconvenient). I just want to get rid of this spurious account as everything with my current Apple ID is working fine.
I hope that makes the situation clearer.