Ahhh, another soon-to-be member of the red mist culture...
I mainly shoot groundhogs. Due to smelling too many roadkills, I will not eat one. My friends will cook up the occasional young one and they say it tastes just like chicken.
Depending on how bad a chuck is blown apart it might stay where it lays. Otherwise, they get tossed into the nearest fencerow. If you shove the carcass back down the hole you lessen the chances that another chuck will move in in the short-term.
What kinds of firepower do you have?