First, a homework assignment. This is generic but you may find something that helps.
Pat’s Big Ol' Mud Page (fixing muddy runs):
https://www.backyardchickens.com/articles/how-to-fix-a-muddy-run-chicken-coop.47807/
Berms and swales may help keep groundwater runoff out of there. Concentrating the water like that can create erosion problems other places but maybe you can deal with that. A cover can help, especially if it is sloped so water runs away from the run instead of in it but rain and snow will blow in from the side. Still, a cover can help.
For thousands of years one way people have farmed hilly land and stopped erosion is the use of terraces. I can't tell enough of the overall lay of your land but maybe a row of cinder block or rock across the bottom of that slope. Fill in behind it, maybe with sand so it drains some, so it is flat. When that flat touches your slope, put another row of cinder block, rock, whatever, and fill flat behind that. If you can anchor it maybe a 2x6, 2x8, or 2x10. Something to contain the fill.
If water washes into to you will still have erosion over the top so stop water from rushing through it. You will probably have to do the sides as well as across the bottom.
They will still scratch and gravity will move fill downhill. You will probably have to top off the fill regularly or at least shovel it back to the top. But from your description and the photos I think terracing is the best bet.
Good luck!