Cut finger tip on a mandolin. should i leave the skin on? or cut it off?

Superficial skin: Injury to his fingertips involving a fin slice of skin that is just hanging by a small thread certainly can be removed however if it's only a partial injury then often stitching is better as it will heal faster sometimes the very edge of a skin flap can't be stitched but provides a little bit of soft tissue protection. The main thing is to make sure there's no bone injury. And avoid infection.

Maybe not: If the piece is very small, just apply a small amount of vaseline from a tube or aquaphor twice a day and it should fall off on its own. Cover it with a band aid.