Here in good old Florida if the mud has a thin layer of water over it you can get threw it with some work. One way is to use a push pole or canoe paddle if neither are handy there is way #2.
#2 ... Providing you have an anchor , take and toss it ahead of the boat and then pull on the anchor line to make a little headway. For some reason the anchor will come back about half the way it was tossed. Don't worry about staying clean because there is no way you will manage that.
I have found that a flat bottom boat ( Pirogue ) is the easiest to travel threw mud with since it is a shallower draft then a canoe or kayak.
One area I like to fish with the pirogue has a border of cattails that are about 60m yards thick at the end of a small canal. I grab a handful of them and pull the boat threw them till I get to the open water to fish it. It is a good sized lake and no one fishes it because it is imposable for the normal boats to get to it. Some great fishing.
With Ron and the guy's on the Brazos ( Texas ) the easiest way to get threw the mud was to walk out threw the mud to the area where the water was over the mud. Then get in the canoe and paddle off down river.
The canoe is sitting in the mud and had to be pushed out a few more yards to a sandy ( less mud ) bottom. The deepest mud was from the bank to the water.