We use a small submersible pump that will pump out mud. It works best if it's muddy water as opposed to just mud, so you may need to keep adding water until the mud is gone if that makes sense. Kind of dilute the mud with additional water and keep pumping.
Ours is made by Simer - great little tool when you have water gardens.
I guess that depends on what you mean by “light mud.” Are you saying that you stirred up the crud on the bottom of the pond and now the water is murky? Or do you mean mud, like you would need a shovel to remove?
Some sump pumps (not meant for continuous use) can handle varying degrees of “dirty” water. And there are some pond/waterfall pumps that are rated as “solids” pumps. Now, as far as which pumps work best, you just need to do some internet research. Just be sure it (whatever pump you get) is rated for “solids-handling,” or “dirty water” use.