Well I applaud your Mum she did the right thing, your snake was injured and she did what any responsible person should do and it is taking the animal to a vet.

I think you should be grateful that she is taking care of your animal during your abscence and that she care enough to seek help for your animal.
the vet said it was a burn and is giving her shots/antibiotics and is costing $185 (my bp cost me $30, captive bred and hatched)
Does this mean that because your animal is 30$ he is not worth spending 185$ in vet cost?

Now no one here can diagnose an animal online obviously your BP has already been diagnosed anyway.
don't see how it can be a burn, because we've had the same setup for 3 months no problem (although I have been helping her shed...taking off areas she couldn't)...it's a 10 gallon glass aquarium, I used to sheets of newspaper, and a UTH
Because you have the same setup for 3 months does not mean the UTH cannot overheat! Do you have a reliable thermostat