I am emailing this to a select group of you who have reported problems backing up to your external hard drive while on Mac OS 10.15 (also known as Catalina).
Catalina hasn’t been a great OS in my opinion and I have tried to keep as many of my client on OS 10.14 as possible. Nevertheless, either because of a feature like (Apple TV) or just because the updates were hard to avoid — you are on OS 10.15.
I was with a 10.15 client yesterday. Her Mac hadn’t backed up since December. She manually clicked on the Time Machine symbol (up by the clock in the top right area) and clicked Back Up and her Mac did indeed back up to the external hard drive. If you have a laptop and do not leave that hard drive connected at night as you might with a desktop, you should manually trigger the backups.
Apple just pushed out an update to Mac OS 10.15.4 on 3/24. Please make sure your Mac is up to date. Click the Apple Menu — top left. System Preferences and Software Update on the 3rd row down. Install the updates, if available. Computer should restart.
Try your backup again.