Adding wheels to it is a great idea to be able to easily move it from place to place to help prevent the ground from becoming what you describe. How many birds to you have and what are the dimensions of your coop and run? The recommended minimum square footage of the coop is 4 square feet per bird and 10 square feet per bird out in the run. You may want to consider adding on to your run to give your flock more space outside which would help prevent such a concentrated mess. It's better to have excessive space than not enough. Cramped quarters also lead to bickering amongst flock members and sometimes even cannibalism.
Adding undyed mulch, shredded or whole leaves, straw, grass clippings, etc. frequently will help also.
