Hamlin Garlin's "Under the Lion's Paw" can be explained by many of the characterizations seen in cartoons. One instance of this is when Scar, from The Lion King, toys with a mouse. He brings his paw down upon the creature, keeping it held captive. Lions, at least in cartoons, can be seen playing with creatures smaller than them, holding the creature captive, and playing with the idea of killing it or letting it go. Essentially, it is a power play.
In Garlin's story, Tim Haskins is under the paw of poverty. Those who possess more than him (land, money, material goods) could be identified as the lion. They, knowingly or not, are oppressing him, much like a lion oppresses a small animal. Here, Haskins can feel the oppression, but he is not "lucky" enough for the oppression to end (through society helping him become richer or by ending his existence).
More specifically, Haskins in under the "paw" of Jim Butler. Butler has rented a farm to Haskins, who has made many improvements. When Haskins speaks to Butler about purchasing the farm, Butler announces that the price of the farm is twice what Haskins thought it was. Challenging Butler, Haskins inquires about the exuberant increase. Butler tells Haskins that the farm is in better condition than when he (Haskins) originally rented it. Therefore, he is under Butler's "paw" when it comes to buying the farm. Butler can charge any price he wishes; the farm will sell to someone, even if it is not Haskins.
In the end, Haskins buys the farm, removing the "paw" of Butler from his life forever.