I have taped a lot of drywall, which has had the same issues as yours does. In the corners, I never filled the gap, I just let the tape span the gap, and mud as usual, being careful while setting the tape, that it not recess itself into the void, making it necessary to use a lot of mud to float it ot to the drywall surface, forcing it to take a long time to dry. Same with gaps between sheets. If it get filled while spreading the mud for taping, good, if not no worries, go as usual.
The only time I find issues while doing this is when the tape gets wet, it wants to swell a bit and buckle out, since it has no mud where the void is. This does not happen with the corners. I pay attention when this happens, and "touch" the tape back in flush, not too hard, it will move easy. If it does stay bulged out a small bit, it will be easy to detect when the next coat goes over it, or not... it may still show. This is where I carefully add another layer well feathered out, so it will sand out well. Sometimes it will still let the sanding burn through to it. Then that takes another tight skim to smooth out the exposed fibers of paper, if that.
The only time I have had issues with "not enough mud" has been with not enough mud on well fitted joints where the base coat was insufficient to wet the back of the tape being applied. It does not take much, but it does need to be enough to at least squeeze a little out. If this does not occur, then when the next coat goes over the tape to build it up, and the tape gets wet in those places, the tape will bubble and the only way to fix it is to cut out the bubble and add new tape.