How can I stop my iPhone displaying "iCloud Backup requires that you verify your password" repeatedly?
I personally had this same message. I turned off iCloud backups, then turned it back on. Settings > iCloud > Backup. While it did ask for my password the first time, it has not asked since.

If that doesn't work, I suggest to again, turn off iCloud backup, delete the backup, then turn it back on.


Settings > iCloud > Backup, turn off.
Settings > iCloud > Storage >
Manage Storage, locate the backup, tap edit in the top corner, and
delete.
Settings > iCloud > Backup, turn on.