yep- you need a winch! and the right tires for riding through slop like that
we have a lot of mud / clay pits like that around here in new england. If I buy an atv and it has no winch- it's the first thing I do
in your situation, wrapping a rope around the rear axle or tire and tying off to a tree would work well to get you out of that mess. if done right in low gear, the axle will act as a winch and pull you right out a little at a time.