Location of pictures file for PHOTOS App on Mac Air

The Pictures folder inside your user Home folder is the default location, where new Photos Libraries will be created, unless you move them somewhere else.

To check, if the Photos Library is included in your Time Machine backup, you may want to enter Time Machine and look what is there.

• Open the Pictures folder in the Finder.
• make sure, your Time Machine drive is mounted.
• With the Pictures folder as the frontmost window enter Time Machine and use the Time Line on the right border of the Time Machine window to go back in time.

Are you seeing your Photos Library in the Time Tunnel in the Pictures folder?

The Time Machine backup of your Photos Library will be complete, unless you are using iCloud Photos for your system photos library and "Optimize Mac Storage" is enabled to save local storage. Time Machine can only backup the originals that have been downloaded to your Mac, not the items that are only in iCloud. See: Restore a library from Time Machine in Photos on Mac – Apple Support (UK)

The pictures should be inside the Photos Library bundle. By default, that would be created inside the Pictures folder in your home directory, but it can be moved or created elsewhere (within limits).

We can tell your Photos Library bundle still exists somewhere, because you still see your pictures and folder/album structure when you launch the Photos app. The question is where it got moved. One possibility is that it was moved to the Trash.

To find out where your currently active library resides, open the Settings panel inside Photos (Photos menu, select item Settings...) and click on the General heading. The first line should say "Library Location:" and the path to the library bundle.