Update

To perform this action will require booting from a >>>> Bootable Installer <<<<.

The Steps below will Wipe the Entire Drive of Everything and there are No Do Overs.

This will have to be performed from a Qualifying Computer to run the version of macOS to be made on the Bootable Installer. Example : Bootable Installer of Big Sur would have to be done on a computer that Qualifies to run Big Sur.

Once that is done read on for preparing the Destination computer  >> Special Notation - works on Intel Based Apple Computers & May on Apple Silicon M1 CPU

Extra Special Notation regarding the Touch ID equipped Apple Computer. About Startup Security Utility and Must Enable from Recovery Mode the ability to boot from External Drive Before Attempting 

1 - Shutdown computer and disconnect all external drive Except the newly created Bootable Installer.

2- Restart and immediately hold the OPTION key until the Startup Manager appears and choose the USB Drive. 

3 - It will present options >> Disk Utilities >> View >> View ALL attached Drives.

4 - Choose the Upper Most Drive ( not the volumes indented and list below ).

5 - The drive normally is called Apple Media or Apple SSD - that is the drive to Erase and format as APFS with the GUID Partition Map. 

6 - Once that is done >> backup out of Disk Utilities and choose install macOS. 

7 - Follow the prompts and it may automatically reboot several time. 

8 - Upon a final reboot - Setup Assist will present with the newer version of macOS.

Shutdown computer my any means possible including the Power Button.

Disconnect all external devices and plug into charger.

Restart the computer in Recovery Mode ( INTEL - Command + r ) immediately at restart. If Apple Silicon M1 >> HERE for instructions.

Once in - Get into Utilities >> View >> View all attached Drives. Run the Repair Disk utility until no error messages are presented.

Exit Disk Utilites and Exit Recovery Mode by a normal reboot and test

User wrote " I updated the software on my MacBook Air " update as in update from Big Sur 11.5 to Big Sur 11.5.2

Or do you mean upGrade from MacBook Air 13″, macOS 10.14 Mojave to Big Sur ??

Judging what is being experienced - the Internal Drive does not have enough Empty Disk Space .

Your only options are Transfer files between two Mac computers using target disk mode and hope you can move enough personal files, videos, pictures, podcasts, iPhone backups etc off of the drive.

That or wipe the Entire drive of everything and do a Clean Installation.

Understand, doing this will remove everything and there are no do overs.