Its hard to tell someone else how to shoot. There is a lot of good information so far, but I will add one more. Turn your scope power down. The higher the power the more it looks to the kid like he is shaky. That will lead to jerking the trigger as the crosshairs go past the target. You will not readily notice a trigger jerk like you do a flinch. The movement may be near imperceptible. When he misses does he pull right, or does he push left and low? Show us a target.
Here is the target at 100 yards. I couldn't tell you which order they were shot in. I can say that on a bench, I was able to shoot with a group of 3 around 1 inch.

He does wear ear protection all of the time. Not only to protect the hearing, but to stop flinching.
I did buy an airgun a couple of years ago just for practice. Unfortunately, even though I bought a scopes made for these types of guns, they keep getting out of wack. The current scope on the airgun is way off.
I think I will have to borrow a .22 with a scope. I don't have one but I know people who do.
And if anybody in the Bismarck area with good shooting experience would like to meet at the gun range to watch him shoot, I would be for that.
Lastly, I don't think I put too much pressure on him to shoot. He likes shooting and hunting and I try to help out without being too critical. I decided to put in for a moose for him because the opportunity is good for moose in North Dakota. 10 years from now, there may be none. I think he will be OK. Just gonna have to keep the shots close.
I'm not saying he can't hit anything. The last two years he has gotten his whitetail does. But they were close shots from a stand. This year, we are going to be walking and so the shooting is going to be more difficult.