A few possible fixes exist for this problem, and each takes only a short time to try.
Eject the external backup drive.
Then reconnect the drive and restart the Mac.
This is the most common solution for the Backup Volume Is Read-Only error.
Instead, follow these additional troubleshooting steps.
Unmount the backup drive.
If the drive cannot be ejected from the desktop, unmount it using Disk Utility.
Turn off Time Machine inSystem Preferences>Time Machineand useDisk Utilityto runFirst Aidon the Time Machine backup drive.
When the process is complete, exit Disk Utility and turn on Time Machine in the System Preferences.
Is It OK to Use a Drive That Needed Repairs for Time Machine?
It’s unlikely that you accidentally set the drive to be read-only.