MUDDY AROUND WATER.....lay some landscape cloth where you want the water bowl to sit, then pour pea gravel over the top and extend it a little beyond that space. Set water bowl on top and the mud problem is solved at the water bowl. The water will run off...where it runs to depends on where you place the water bowl. I place mine next to the fence so the water runs out of the run. Or, try another BYCer's idea and lay an area in fake grass carpet in the spot under the water bowl. Thought that was a great idea also. However, make sure there are no loose ends on the fake grass carpet as the ducks will pull on loose ends and eat it. Then you'll have another problem if that happens. You have to have something under the water bowl for the water to run off or you will have one big muddy mess.
My whole run is covered in pea gravel, over landscape cloth, over 1/2 inch hardware cloth stapled to the runs wooden frame. My ducks do not eat the pea gravel...they have food & water available in the run 24/7, plus oyster shell in a bowl to munch on at their leisure.
When my ducks are out of the run and outside in their kennels, yes, mud is a problem where the additional water bowl is placed. Cover the area around the waterer with straw to keep the area as dry as possible. They love to make holes in the mud, so beware and don't step into them and twist an ankle. Ha! Ha! Seriously, it can happen. Hope these suggestions help you with your mud problem.