The first thing would be to modify your run so that you can get into it.
I made the mistake of having a half-high run in my first iteration and have since determined that the extra lumber and hardware cloth is cheaper than the chiropractor visits.
Unless you've got an odor or mold problem you don't actually have to clean out everything, just harvest the compost/mulch you want for your landscape and bring the level down so it's not spilling through the wire.
Coarse wood chips are often considered the gold standard for controlling mud and odor. Since you're in the US northeast you can expect to have a bonanza of fall leaves soon -- they make a great addition that chickens absolutely LOVE. But they *can* pack and mat if not mixed with other materials to help them stay loose and fluffy. Many people store bags of dry leaves to use all winter.