Hi Lottie51
Before erasing the external drive there is one extra check you can do that might resolve the problem.
Open Finder, locate the external disk in the list of drives, mouse-right-click on the disk and from the pop-up context menu click "Get Info"
An Info pane will open, at the bottom of the info pane look under Sharing and Permissions.
To be able to write and read to the disk you should see (normally) three or more users, "your own name", "staff" and "everyone"
Occasionally, when upgrading a macOS the users and permissions may become scrambled.
Against "your own name" you should have Read & Write Privilege, "staff", if present, should also be Read & Write, finally, "everyone" is normally "Read only".
To be able to use the disk for a Time Machine backup you must have Read & Write privilege.
If your name is missing from the list of users, or you only have "Read only" privilege, click the lock symbol at the lower right of the info pane and enter your user password, then, if your name is already in the list click the privilege box next to your name and change the Privilege setting to Read & Write.
If your name is missing from the list of users click the "+" symbol at the bottom of the info pane and a pop-up window will open, look for your name in the list , click it to choose and then click the "Select" button.
The final step is to apply any changes you make to users or privilege settings.
To the right of the "+ -" boxes is a circle with three dots, click that circle and a pop-up menu appears, click "Apply to enclosed items"
The revised permissions and ownership will now be written to the disk itself, and all the contents.
This will take a long time, several hours if the disk has a lot of small files and folders stored.
After making any changes it is best to leave the Mac alone while this change propagates across the disk, for a 1TB drive it might take four hours to complete.
When the permissions have been updated, reboot the Mac and then try Time Machine again to see if it can now write to disk.
An extra tip, when looking at the list of users and permissions if you see under users, "unknown" or "searching" this is an indication that permissions have been set for a user that can not be identified. If this is the case, reboot the Mac into Recovery-Mode, open Disk Utilities and use First Aid on all your Mac internal and external disks so that the correct user can be identified and this error repaired.
Extra tip2, when applying First Aid from Disk Utilities it must be applied in the correct sequence, otherwise it will fail to repair permission and missing user problems.
Expand the "tree" view for each individual disk to view the nodes and always apply First Aid from the bottom up, Volumes first, at the bottom of the tree, then Containers, in the middle, and finally Disks, at the top.
If you start with Disks, for those disks that do have Volumes, Containers and Disks, user permissions are not repaired for any of the nodes below "Disks".
https://support.apple.com/en-us/HT210898
HTH
Will.