Use USG's Durabond. THis is a hot mud powder that you mix. If you push Durabond into that seam, it will become harder than the drywall and also will permanently glue it in place. Just make sure that you scrape any excess off before mudding and taping.I didn’t place this one stud far out enough to catch screws so it’s a little loose. Maybe 1/16” of play on the drywall if push on it.
Is there a type of mud that will firm that corner up?