Are there good treatments for cuts in the mouth that won't interefere with my mouthguard?

Cuts in the mouth. All depends on where the cuts are coming from and why are they not healing. In the mean time just go to your local pharmacy and ask the pharmacist for over the counter gels to help with the cuts and if you don't see am improvement in 2-3 weeks you really need to see a dentist to figure out where the cuts are coming from.
Cuts in mouth. Yes but it all depends on why you are getting the cuts in your mouth. For example are you wearing braces on your teeth? Depending on the cause, consult with your doctor about treatment.
Yes. Depending on the cause, severity and location of the cuts, therfe are some ointments that can be placed on the cuts. There is an rx called kenalog (triamcinolone) in orabase which works well on cuts and some other one's that are otc. Sometimes the mg can be cut back or the ointments can be placed inside the mg.