A H&R single-shot in 20ga has potential to kick like a mule, just be aware. They only weigh about 6lbs and target shooting with one could be a painful experience. As Cartwheel said, find featherlight loads or he'll likely never want go to the range again.
I'm 6' 1", 250 pounds and even I don't enjoy shooting my friends H&R 20. I found it to be a real face slapper the times I've shot it.
Ouch.......
Just shot the .20.......
No way my kid d can shoot it and enjoy himself.
I wouldn't be able to shoot it for a couple rounds.
Back to the drawing board.
I hate to say we told you so but the best advice handed out here was stay away from a lightweight single!! They hurt and that is no way to start a kid off. He'll probably be scared to pull the trigger on the next shotgun and pull his head away from the stock and close his eyes no matter how soft a gun you get him. I shot a 16 ga SxS one time as an adult that was so brutal in recoil it was insane and I can handle recoil but much prefer not to. As to your question a plain barrel will work fine but you seem to be wanting to go down the road of something not really designed for the task?