Hi - iCloud is a syncing service, not a backup service. It is designed to sync your library across all your devices. The consequnce of this is that normally all your photos are stored on your devices AS WELL as in iCloud. You must also have enough iCloud space for your entire library. The included free 5GB is rarely enough. Are you paying for an iCloud plan?
It is also very important to understand that if you delete a photo from one of your devices (including your MBA) they are deleted from iCloud and all other devices.
There are two alternative ways to reduce the space taken by photos on your Air.
1 - use iCloud and "optimise mac storage" (one of the iCloud settings in photos). This will store full size masters only in iCloud and replace them on your mac with much smaller preview files. Full size images are downloaded when needed for editing or output from the library (printing/export etc). The main issue with using "optimise mac storage" is that backups now become problematic, because you don't have the masters on your mac, they are not included in your backup. See:
https://discussions.apple.com/docs/DOC-250001961
2 - Store your library on an external drive. You can still sync it with iCloud if you wish. If you were to go that way, then to correctly prepare the external drive:
It must be formatted APFS or MacOS extended (journaled) (preferred for spinning drives)
It Must not be (or have been) used for time machine. (A time machine drive can be used if re-formatted)
It must have the 'ignore ownership" checkbox ticked.
It should not be on a memory stick, SD or similar.
It must also not be a network drive (a direct connection is required - eg USB or Thunderbolt).
File sharing services such as iCloud Drive, Drop box, Onedrive etc are not compatible with Photos libraries.
It stands to reason that if you want any sort of reasonable performance it should be a fast drive with a fast connection.
It must also not be a network drive (a direct connection is required - eg USB or Thunderbolt). File sharing services such as iCloud Drive, Drop box, Onedrive etc are not compatible with Photos libraries.
See
https://support.apple.com/en-gb/HT201517