Instead of risking cutting your finger a bit of cotton works well on inside of tires instead
of your finger. Where it snags is where you probably have bits of glass/wire etc. penetrating
tyre. If you removed all the glass, and its a small cut (not more than a few mm long)
the tire should be ok to ride. If its sliced, get a new tire. Some people use shoe goo
(used to repair running shoes) to fix cuts in tires.

Marty