They are all over the place and, IMO, not worth the time or money you can expend trying to get them to shoot. If the have the ejector system, don't go anywhere without a way to knock out a stuck case.
My exp with them includes two .223, two 30-30/12 GA combos and a .308. The 30-30s were fun and delivered ok at 50 to 100 yards but not any better than a typical '94 or 336. The .308 was suitable for deer or elk out to a max of 150 yards at best.
The .223s were the biggest disappointment. Four inch and larger groups at 100 yards on their best days.
I eventually got a Tikka T3 lite in .223 at a cost of approximately $500. Compared to the cost of a Handi and the wasted loading components, $500 is cheap.
As mentioned above, you can read a whole lot about them at the Greybeard site. They are huge fans of the Handi over there, but when you visit there ask yourself why the discussions are mostly about how to cure accuracy problems.
Get your kid a Quality bolt gun.