Led pot light wire damaged at rough in by drywall cutout tool. How to repair?
My non-code, unofficial, somewhat hackish, minimally risky solution would be to:


Cut the wire as long as possible near the socket base. You'll need at least 3/4" of good wire. 
Cut the wire at an undamaged point on the other side of the damage. 
Strip both wires 3/8" and twist well (but not so vigorously that you damage it).
Install a butt-splice connector of appropriate size with a good crimp. 
Wrap the repair with tape, then wrap the other wire in with it to create a splint to prevent pullout. Use a high-heat tape as this is a tight space with heat-resistant insulation on the wires.