Bathroom cabinet door fix
Wow! That is unfortunate.

The material is particle board and has been badly damaged. Since the cabinet is against the wall repairing it would be very difficult without removing the cabinet first.

I would try to move the hinge. Preferably, to the top of the door if you have enough space for the hinge. If not move it below the damaged area. You may have to drill very small, maybe 1/16", shallow pilot holes for the screws to get them started but you could try driving the screws without this first.

Have the 13 year old help you so he/she can learn to fix their mistakes and how much work it takes. 

Good,luck!