Time Machine could not create a backup because the computer was running on battery power. Connect your Mac to a power source while your backup volume is available. Backup tells this every time, although my MAC is connected...?!

Additionally, give this a try: boot into Safe Mode according to How to use safe mode on your Mac and test to see if the problem persists. Reboot normally and test again.

NOTE 1: Safe Mode boot can take up to 3 - 5 minutes as it's doing the following; 

• Verifies your startup disk and attempts to repair directory issues, if needed

• Loads only required kernel extensions (prevents 3rd party kernel/extensions from loading)

• Prevents Startup Items and Login Items from opening automatically

• Disables user-installed fonts 

• Deletes font caches, kernel cache, and other system cache files

NOTE 2: if you have a wireless keyboard with rechargeable batteries connect it with its charging cable before booting into Safe Mode. This makes it act as a wired keyboard as will insure a successful boot into Safe Mode.

Also check the battery icon on the menubar to see whether it shows a charging symbol or a power plug symbol to indicate the charger is recognized by macOS. Don't assume the laptop is using the charger just because the charging cable is connected.

If you’re at 13.1 as per the footer here, update to macOS 13.2.1, and try again.