This is a MacBook Air and/or Mac operating system issue.
On the MacBook Air......
Click on the Time Machine "clock" icon at the top of the Mac's screen
Click on Open Time Machine Preferences
Click on the Options button at the lower right of the window
If the box next to "Back up while on battery power" is unchecked, enter a check mark in the box and then click on the Save button
If the box next to "Back up while on battery power is checked, remove the check mark from the box and then click on the Save button
Close the windows
Restart the MacBook Air
Click on the Time Machine clock icon at the top of the Mac's screen
Click on Open Time Machine Preferences
Click on the Options button
If you want the Mac to back up while on battery power....and the box is not already checked......enter a check mark in the box next to "Back up while on battery power".
Click Save
If you do not want the Mac to back up while on battery power......and......the box is already checked, remove the check mark from the box next to "Back up while on battery power"
Click Save
Close the windows and restart the Mac again
Check to see if backups will return to normal
The actions above will reset the Time Machine preferences on your Mac.
If you continue to see the same "Power source - Power adapter" error message, and no other suggestions will work, you may need to reinstall the Mac operating system, since Time Machine is part of the operating system.