Never Had One But Want One

Hard to get a break barrel to take medium game and yet have light recoil. Not sure what to recommend that would fit your criteria. If by medium game you mean rabbits , hares, etc. I would recommend a Diana 54 Air King. Powerful but not a break barrel. Lighter recoil as well. The Sig ASP20 fits the bill but it's got some kick. Cocking is very easy for its power.
 
Hard to get a break barrel to take medium game and yet have light recoil. Not sure what to recommend that would fit your criteria.



Can't believe I'd say such a thing, but perhaps forgo the airgun in that case brawler. 

One can get a very fine Ruger 10/22 (or similar) .22LR powder-burner for ~half your budget. And arguably less recoil. :)

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Hard to get a break barrel to take medium game and yet have light recoil. Not sure what to recommend that would fit your criteria.



Can't believe I'd say such a thing, but perhaps forgo the airgun in that case brawler. 

One can get a very fine Ruger 10/22 (or similar) .22LR powder-burner for ~half your budget. And arguably less recoil. :)

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Problem with powder burners is noise and over penetration. Great picture! 
 
Depends on the range of your game. I have seen coyote shot with a 177 springer (head shot) DRT, he was so interested in the chickens in the coupe he didnt notice the man 20 yards away. I own a .22 HW95 that I have no doubt 50yards and under would take a raccoon or other game of that size with proper shot placement. Not terribly hard to cock but if I really wanted a springer dedicated to medium size game I would get a RWS model 56 in .22. 
 
Sig asp20 is a good bet. 23FPE and accurate as can be. Recoil not horrible. I have had my daughter shoot it with no issues. 2 shot 20 yard group below (first time she shot it, after hole in hole I told her to let me take a pic asap)
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I think cocking effort on mine is near 33 pounds. Trigger breaks over 2 lbs, but feels lighter than that and is very predictable with no creep in mine.
 
My break barrels in .20, .22 and .25 have easily taken both raccoons and woodchucks out to the 30 yards or so I have to shoot in so yes, it can be done. Between the eyes or between the eye and ear has taken every one of them I've ever laid crosshairs on. 

Recoil, which you asked to have be minimal, is where the sticky part comes in because a springer, regardless of caliber has it. Shooting one and accurately takes practice. There's also 35-45 pounds effort needed to cock one, so also something to take into account and maybe why Jerry said a break barrel was not the correct tool?? 

The only way you're getting a minimal or no recoiling air rifle is CO2 or a PCP. 
 
I say the Sig ASP20 in .22 would be perfect! Great guns and for a magnum powered break barrel thought it was pretty tame and easy to shoot accurately.

It is a great gun. I had a large learning curve with respect to holding it and grouping well. Once I understood how to replicate my hold, it became an easy gun to group with.
 
 I sit along the lines of seven08. There is no recoil to my physical body to worry about with an air rifle The only recoil I worry about is the recoil that hurts the scope or shakes the rifle apart. If thinking of youth type shooters I'd be thinking about the (Cocking Effort). The model 34 in RWS,Diana at 33 lbs of cocking effort is fairly standard.Or Weihrauch 95s which I'd bet is pretty similar and come in 177 or 22cal. These rifles can also be "detuned " by installing a less powerful spring and then re-installing the original spring when ready or bought with a european tune of 12 footpounds of hitting power from the muzzle, to lighten the cocking effort. I can only see this done for a very young child 6 to 10 yrs or for personel preference. Any 12 year old from my day would develope a technique. It's mostly toning up the set of muscles used in cocking. Once they're toned up it's easy. The 34s r the way to go. Step up with the Weihrauch 95