I like deep litter in the run to keep it dry and clean.
The bedding -- a mix of wood chips, wood shavings, pine straw, straw, fall leaves, and whatever else I have handy -- absorbs the poop and neutralizes it in a cold composting process while keeping the chickens' feet dry and clean (important for me because I like Brahmas and other feather-footed breeds). Putting down layers of the above would go a long way toward solving your mud problem.
Also, if you have drainage issues with ground water running into or collecting in your chicken run you should consider using grass swales and/or French drains to correct the drainage before long-term health problems occur.