Well a muddy run can lead to bumblefoot and chickens scratch after laying an egg. So I would think that if your run is very muddy then it’s getting into the nests and onto the eggs. Resulting in dirty nests and eggs. These can be washed but it’s much easier to deal with the run than to continue to wash eggs.
When all of the snow melted here I had a problem with my run being very muddy also. I solved the issue by putting down pea gravel and mulch. The mulch stays dry, since it’s sitting on a layer of gravel, while the dirt under is what gets wet. I also covered part of my run.