If it were me, I would remove it from the current enclosure and keep it in a sterile environment until the nose heals. Dirt substrate, fecal matter, etc. can harbor bacteria which could lead to infection. A shoe box with a paper towel or newspaper substrate, kept spotlessly clean, would minimize that risk. The gecko should be kept warm and well fed during that time, and should have access to a humid hide.
It appears to be rubbing the nose on something abrasive in the enclosure. It is important to figure out what that is.
For treatment, I would swab it with betadine, and then apply an antibiotic ointment periodically. With diligence, it should heal.