Maybe. There are two goals in repairing a damaged wire:

1. Maintain the connection. The wire is stranded, so if half of the strands are broken, then only half of the electrical load (amps) can be supported. Fix this by replacing with a solder, crimp, or a piece of new wire of same gauge.

2. Insulate the conductor. Pick your poison. My favorite is shrink tube (get it anywhere, home depot, radio shack, etc). It goes over the wire and repaired area, and shrinks to seal when heated with a heat gun (or hairdryer maybe...). For a very small nick in the insulation, silicone sealant works well. You may have to put on more than one application. Electrical tape has a temporary characteristic. It gets brittle when weathered, and the adhesive gets messy over time.

Nate