Backing up to external drive

my last successful backup was in Oct. Now when I try it it fails and tells me that "Time Machine couldn't complete the back up to " ". Because the Macintosh HD is nearly full. So I deleted whatever I could and got just enough space to back up to " ". Now it will fail the next time yet I have 5TB on my external backup drive. So I don't know what to do. I do not want to do iCloud storage and I thought by backing up to my external drive I would never have an issue. Seriously I don't have that much stuff on my computer. Now I see that under General--Time Machine--Options that my external is set to be excluded from backups. So I am totally confused.

Any advice on how to solve this problem? 2023 MacMini...running Sonoma. Thanks