Time Machine back-ups fail ever since upgrading to Catalina OS. Any solutions?

Have you held the option key down while clicking on time machine (in menu bar at top. if not there go into system prefs -> Time Machine and check button to put it there) and done a "Verify Backups"? It is possible that the file is corrupt and you will need to do a start a new backup.

My Passport appears to be a USB connected external drive.

Please supply screenshots (obscuring personal information) of Disk Utility when the drive is plugged in, and of the error(s) you receive when attempting to back up to it

Also, do attempt this (as suggested above)

Have you held the option key down while clicking on time machine (in menu bar at top. if not there go into system prefs -> Time Machine and check button to put it there) and done a "Verify Backups"? It is possible that the file is corrupt and you will need to do a start a new backup.

Screenshot any results of this, too

Please explain what you back up to, and how it is configured