Unable to back up my MacBook Air through Time Machine

If Time Machine cannot create a local snapshot on your internal drive (for backup purposes), your filesystem could potentially be corrupt. To repair your filesystem (assuming macOS Catalina 10.15.x):

1. Start up from macOS Recovery.
2. If prompted for an administrator password, enter it to continue.
3. When the list of utilities appears, select Disk Utility.
4. In Disk Utility, click View -> Show All Devices.
5. Locate the APFS Container entry in the left sidebar. For each volume indented below it:
   1. Select the volume.
   2. If a Mount button appears, click on it. Enter your login password if prompted.
6. Select the APFS Container and click First Aid.
7. Wait for the First Aid process to complete.
8. If any errors appear, click the arrow next to "Show Details", and take a screenshot of any error messages that appear in the log. Otherwise, if the operation is successful, restart your Mac and try backing up with Time Machine again.

What is the error that you're encountering?