Filling in drywall gaps
First, know that drywall tape joints are actually quite strong. If you leave a 1/8" gap between panels so that joint compound can achieve full depth, there's little you need to do in that location. There will be virtually no pressure on it there. 

I'd float some scrap lumber between the studs in the middle to reduce the span by half, and maybe at the upper left. 4" lengths of 1x2 or plywood will do. 

You could also use the hemmed edge of a drywall sheet for even more strength. It doesn't look like the wall was furred out for the flange, so I'm not sure how things will fit, but doing so might benefit by it being a thinner piece.