Pop up about iCloud popping up

Hi, my husband had a problem with his Apple account - he was locked out of his accountt. With help from the Apple support team he got a new account and even managed to get some of his data back from his old account. But he cannot access his old iCloud account and he wants his photos and some vital info from that. Now a new problem has arisen. There is a box that keeps popping up asking him to sign into his iCloud account. He tries to - but it comes back with ‘verification failed’ message. Now this is happening every 2 minutes or so. He can’t use his laptop for anything cos of these pop ups

We have spent so much time, effort and money in phone calls to the Apple support team with no joy.

help please