LoloR, I am BRAND NEW to this, but also have a covered house and run on a heavy clay soil, with a lot of rainfall (looking out the window, i see the gauge is still filling). I'm using straw in the house, the chickens perch there, but the ducks bed downstairs by choice. Their scratching ensures a moderate amount of mostly clean, mostly dry, straw falls "downstairs", where it helps to keep the ducks warm and dry. The house itself is in the form of a U, so I can roll the wheelbarrow in and rake spent straw directly in, rather than shovelling and lifting, as I'm the lazy sort.
Underneath, I laid a bed of river stones around the watering cups and automatic dog feeder I use to supply the ducks with water, its helped to keep the water clean("er" - they are ducks, can't expect too much!) and their feet out of the clay and mud when they splash.
For much of the run, I lightly tilled the dirt and incorporated cheap bird seed and clover before they were moved in. Provides a soft carpet of mixed greens and the occasional grain seed to perpetuate it when a plant survives that long. Also improving the soil nutrients and water holding capacity, attracts insects (mostly grasshoppers and crickets) on which they feed, and slows erosion from runoff. I tried adding peas and green beans to the outside of the fence around the run as well, but a very heavy rain days after sprouting, plus the scratching of the hens when free ranging has rendered that experiment almost a complete failure.
Won't claim its best, but it is (for now, at least) working.