How can I repair a nicked leather strop?

Small nicks can be sanded flush with a good quality sandpaper of 300-600 grit. It is not advisable to use a pumice stone since it can leave grit in the leather.

If the cut resulted in a small loose flap, first glue the flap with cyano-acrilate (CA) glue ("wonder/super glue"). To do this, apply a very small amount at the bottom of the nick, press it firmly together, and wipe off any excessive glue. You will have to be very fast since CA glue sets very quickly, then sand smooth as above.

While not esthetically pleasing, small nicks do not compromise the performance of a strop.