If you are a reloader or have a friend that reloads you can make some low recoil loads. Don't get an ultra light gun action type would be your call. My sons and daughters started with a 37 Ithaca 12 gauge with reduced loads. I cut 1" off the stock and put a good recoil pad on it. I knew it was throwing around 60% and POA was same as POI.
I like the idea of a single shot as others have said the problem is generally the guns are so light. What do you guys think of making a plug for a pump or auto that blocks the whole magazine making it a single shot. It would be quite easy to do.
The Mossberg 500 bantam I mentioned earlier comes from the factory with a full length plug in the magazine, limiting it to a single shot. The stock dimensions are also meant for kids in every way. Unlike a cut down adult stock, the pistol grip is smaller in circumference and positioned closer to the trigger so a small hand can get a good grip and small fingers don't have to stretch to reach the trigger. The forearm is also positioned closer to the receiver so short arms don't have to stretch to work the action. All in all, a much better choice for a small child than any cut down adult sized gun and better than most other youth guns.
