OK... .410 is small for deer in any state but it is illegal in IL anyway.

The 870 in my opinion may kick a bit much with that small of a child, for one any pump kicks more than an auto of same weight. Not to mention that children can't always pump the gun or reach the pump properly..

The single shot H&R is a good gun and very accurate BUT, it weighs a ton.. That helps with recoil but it will be tough for a child to hold.. And I know some say they don't kick but I have shot many of them and I'll tellin ya that the single shot really kicks also, no place for the gas to go except out the end of the bbl.

There are many options and working for a gun shop I get this problem quite often with youths and hunting..

In your situation here is my humble opinion..

I would buy a used remington 1100 in 20 ga. IF you want to shoot sabots order a fully rifled slug bbl for a couple hundred bucks, I would choose iron sights to start.

With an 1100 you get lighter recoil, reliable classic gun along with a wood stock that you can trim an inch or so off of for a youth and very easily put a full size stock back on it later when the child out grows the smaller stock. Stocks are easy to find, I've bought several off Ebay for 30.00 or so.

This way you have a gun that can be used for more than just deer, a reliable gun, less recoil, easy to change as the child grows, and really inexpensive for what you get in my opinion..

Just my thoughts, good luck...