Is it an issue worth having fixed? Myself I've had a couple hinge action single shotguns that were like this, should he be concerned?
H&R would fix a factory fitted barrel that's loose, break barrel firearms shouldn't be loose when the action is closed, if they are, they weren't fitted right to begin with or they've been abused and not properly lubricated, a simple shim will fix it and work fine on a shotgun, they aren't too critical, I've got an old H&R that has no serial number, it's been loose without the brass shim I put in it almost 50yrs ago, it shoots fine. I've shimmed many high pressure rifle barrels too, but do a lot more work to fit the shim and make it permanent, see the barrel fitting info in the FAQs.
Tim