iPhone keeps asking for password
When I'm trying to use my iPhone I'm constantly interrupted by popups saying "Sign in to iCloud - Enter the password for [my Apple ID]" or "Apple ID Verification - Enter the password for [my Apple ID] in Settings."
When I go to Settings it says "Update Apple ID Settings - Some account services require you to sign in again". When I do this, I get an error "Verification Failed - An unknown error occurred"
I'm definitely entering the correct password, because it works on icloud.com in a browser.
How do I fix this?
iPhone SE, iOS 11.3