Hello Monarch_1,
Thank you for using Apple Support Communities. It's my understanding that the desktop picture unexpectedly changes after restarting your MacBook Air. You've come to the right place! I'm happy to help.
It is expected behavior for the Catalina island image to appear on the login screen after startup, this cannot be changed. If your desktop picture doesn't appear after logging in, please review the resource below to make sure you've followed all steps to change your background:
Customize the desktop picture on your Mac
If the behavior continues, try testing the issue in safe mode. Starting in safe mode allows your Mac to run certain checks, remove caches, and prevent login items from automatically launching. This can isolate possible third party issues. Use the steps and information in the link below for assistance with safe mode:
How to use safe mode on your Mac
If the background still won't save, test in another user account. This will help determine if the issue is user specific or system wide.
How to test an issue in another user account on your Mac
Let us know the results. Please include specific error messages and any other troubleshooting steps that you may have tried.
Have a nice day!