No it would not.
The antibacterial properties of honey are slim and would do nothing for an internal infection. You can buy some aquarium antibiotic water additives like Turtlefix and add a small amount to his water. It will mostly just assist in preventing external infections. If he was really in pain he would not eat and especially not if he had damage in his mouth.
You may have nothing to worry about at all. Perhaps what came out of his nose were cricket guts combined with hemolymph.