First, you need to reinforce your coop/run. Since it is digging under try installing an apron. Take maybe 18” to 24” of wire and lay that outside your run, then attach it to the bottom of the run. You don’t have to bury it, just lay something on it to hold it down until the grass grows through it, but it’s pretty common for people to remove the sod and put that back on top. That gets it out of the way of lawn mowers and weed whackers. The idea is that the predator goes up to the fence and starts to dig. When they hit the fencing they don’t know to back up. Youj do need ot attach it well to the bottom of the run so the predator can’t squeeze through an opening between the fence and apron.
Since it is happening at night, can you lock them in a safe coop? That’s the approach I use, a predator resistant run during the day and a predator proof coop at night.
That description doesn’t perfectly describe normal behavior of any digging predator I can think of, but all predators don’t read the book and act like they are “supposed” to. If your birds are fairly small, the list of possible predators gets even larger.
It would help to identify the predator. Can you see any tracks or droppings? If it is staying around to eat the bird in the run, there should be droppings. Tracks can be harder to find but it may have left some in the fresh dirt where it dug. It will be back. You can maybe clear off an area and rake the dirt to give a good place for it to leave tracks or spread some flour to maybe get even better tracks.
My first guess is fox or bobcat, though the fox would normally carry it away to eat it. Bobcats often do too but also tend to bury remains so they can come back to eat those later. A raccoon normally doesn’t consume everything like that but eats certain parts. If it is still hungry, it goes to get another. A possum or skunk normally won’t consume everything unless he bird is fairly small. It’s impossible to be sure what it is from your post. They don’t all act like they are supposed to. Until you know what it is, you can’t be sure you are using the right bait or right trapping technique.
Good luck. These things can be really frustrating.