Can I connect an extra hard drive to my MacBook Air?

yes, if you are running low on space, you can purchase an external HDD or SSD of sufficient capacity and move your larger files (videos, pictures, documents, etc...) to it. once they are on the EHD, you can simply erase the same files from your internal drive. and, if it's just going to be used for storage, you can use an HDD and save a few bucks over the cost of an SSD. FWIW, i personally keep very little on my internal drive.

additionally, you want to keep ~ 20% of your local drive as free space for proper operation of macOS.

but please make sure to NOT use any software that supposedly makes the external drive "Mac ready". when you use software from just about any manufacturer for their storage device you run the risk of that same software not being compatible with macOS updates / upgrades. it can cause the software to stop working. at best this will cause you a headache. at worst it can render your data inaccessible. it is always best to use disk utility to erase and format a drive before use to remove any preloaded software.

please see Erase and reformat a storage device in Disk Utility on Mac - Apple Support