Depending on the age / size of the child, I would say Air Arms s200, for smaller child, AAS400 carbine for the slightly older/larger child. Forget springers for ft or hft as they require a lot more skill to shoot accurately at typical targets/distances, especially cheap springers. Scopes I would suggest a Hawke sr6 with as big a mag as possible, Nikko nighteaters 8-32x or 6-24 or target masters should do the job. A combination of any of these will give a decent chance to any kid to acheive decent scores. Of course second hand brings in older big Nikko scopes, the Air Arms MPR, or even a Hammerli AR20. The advice about goin to your local club is very good. All clubs are full of enthusiasts who will probably be able to let said youngster try out most of the kit mentioned in this post.
EV2 mk2 with 2009 big Nikko
HW97K with 6-24 Nikko Nighteater