Filling a hole made in a wood stud
It depends on what loads the stud carry and the spacing. by code you can notch out 25% of a structural member or drill a hole in the center 40% for a bearing wall and 60% for non load bearing. But to be safe you can add  a plywood plate to the edge of the studs and put in about 15 wood screws. This will help carry bending and compression loads.

Inspector Jeff

ICC/FL/SC licensed combination building inspector and building designer