How to cut drywall over stud?
You want to add backing to the inside sides of the existing studs as well as to the centre of the span. The simplest way to do this is to cut the new backing (preferably 2x4, or whatever the existing stud dimensions are) about 4" longer than the height of the opening and screw them to the existing studs. The cut one more piece of backing to the same length as the other two, but this time place it in the centre of the opening and screw through the top and bottom of the existing drywall to attach it. See the picture below for reference.



This method will be much easier than trying to score the existing drywall back on the existing studs and will provide a solid patch.