How to fill a hole in the wall?

I attached a baby stairgate to the wall and ad it's a pressure gate it cracked the wall and went through it so there is a bug hole now - as it's not a solid wall I can see the stud work and the bathroom wall, is there anything I can fill it with as the gate needs to be attached, I was thinking foam filler?

you need to repair that hole?

cut a longer slot in the plasterboard and then a piece of timber gripfilled to the bathroom wall will spread the load and thenreplace the plasterboard and fill / skim / paint etc to restore the wall.

are you fitting the gate to the spot where the hole is?
if so you need to get some timber support behind the plasterboards to help carry the gate. then repair the hole.
Then you can fix the plasterboard on to the timber, rake out about 50mm of the plaster finish on the walls around the perimeter of the hole (that is if it has been plaster finished and not joint filled).
Then apply joint tapes to joints and fill out, you can use multi or board finish for this, or if the hole is not too big you can apply easifill.
Easifill will need two or three applications and sanding down after each application, to smooth.
Easifill can also be used if the method of wall finish was to to joint fill, rather than plaster.

If you don't want to put the stairgate back but just need a cosmetic repair, you can patch reasonably small holes by sticking hardboard to the back of the surrounding plasterboard with PVA. A small hole through the middle of the hardboard allows a string to be threaded through to hold it in place until the PVA sets. Prime the edges of the hole with dilute PVA and immediately fill with Easi-fill.