Fixable or gotta replace?
hi Aloha1, in all seriousness, you can fix this without replacing sheet. I doubt very much if you can pre fill and tape it? as there is nothing to hold the mud to keep if from falling through. short of filling entire cavity.
I'm assuming the wooden beam is to remain exposed? how I've done in the past, is plug patch it. you say it's a 1" gap? I'd cut a 2 1/2" piece of drywall the length of gap from ceiling down to under beam. on the back side of the 2 1/2" piece use sharp knife and score the paper the 1" just the paper! peel paper off, then start scoring gypsum, do slowly until all the gypsum is removed with Spackle knife and only the front paper is left. you want it as smooth as possible, leaving you with 1 1/2" of paper to tape on the plug,
before riding it on, put a piece of duct tape on the beam, down side and bottom of beam, slightly behind drywall line. use a spatula and coat the duct tape with vaseline, the mud wont stick to it much if at all.
then, with your mud, butter it up on all sides of the plug like you would a brick, put a screw in the 2x plate that's showing to help hold the mud. mud all the edges good, place the plug in on an angle, starting against the beam, mud the back of plug enough to allow a bit of mud to slide behind the sheet with gap. mud the sheep with mud enough to set your 1 1/2" paper, use 6" knife to push the plug back into proper position. should make it pretty straight. take you time to make so!
let dry 24 hours at least, finish and feather as needed.
do same with corner bead, leave duct tape in until done and sanded. when all is done, use sharp razor knife, run it along the corners cutting duct tape and remove.
you should end up with a nice clean corner and not a bunch of mud on exposed beam maybe a small bead of caulk
as always, just my thoughts
take what helps? ignore the rest.