How to check if iPhone has been backed to iCloud? A relative of mine has unresponsive iPhone that contains some important data. The phone has not been backed up in iTunes. Is it possible to check whether it was backed up to iTunes, and when?

icloud.com doesn't seem to provide that information. iTunes normally provides that information, but requires unlocking iPhone which is impossible because of unresponsive screen. Is there any other option?
I'm sorry to hear about the broken phone.

Checking for iCloud backups:


iCloud.com will at least show you if there are any backups, but yes unfortunately no info about the date. The existence would be under Account Settings, and then seeing space taken by Backup.

If someone has a Mac, you can create a separate user account on that computer, and then set it up as your relative, using their iCloud credentials. (Just be sure to use a password they know AND delete the account/sign them out/remove the device from their iCloud after you're done so it doesn't become a source for a secondary password for them to set up iCloud later.) Once you have the macOS account, you can see the backups in the manage iCloud storage section of system preferences. https://support.apple.com/guide/mac-help/manage-icloud-storage-mh36833/10.15/mac/10.15 



Checking for iTunes/local backups:


Open iTunes, and select Edit > Preferences (Windows) or iTunes > Preferences (macOS). Under the devices tab, you'll see a list of the backups.
On macOS Catalina, connect any iPhone/iPad, just so you can get a Finder window and select a phone. Then click manage backups, and you'll see a list of backups on that Mac.