Yes, Covered with a roof.
We have a 10'x50' run. It is directly behind the 50' barn, so the run is completely protected from the west winds in the winter. The barn (pole building) has a loft so the barn roof edge is pretty high. We put in 4x4 posts (anchored 2.5-3 feet below the surface with concrete) and created a single slant roof that goes just under the barn eaves, and slopes to the edge of the 10' run width. The materials used for the roof are plywood and shingles for the coop, then metal panels for the exterior/ non-coop portion, then clear panels where there is a window into the barn. The coop is elevated, so they have that covered area as well. It takes time. We are in year 3 of chickens. First we built a brooder, then a coop, then a 6' roof over the decking (bc it's elevated) and door into coop. Covering the area around the door is a primary concern if you get snow and ice in the winter. Last summer we installed the posts for the rest of the roof, and started to add supports for the panels and we bought the panels. We put up a few panels during ok winter days. We still have another 10-15' to go to fully cover the 50'run. But, it is a great thing to have more dry ground in the run.
The first year we had a very heavy rain, and water flowed through the south portion of the run (partly bc the barn downspouts emptied on that side) and we were thankful the coop was elevated. Water didn't stay around, but it did wash out the bedding in that side of the run, resulting in annoying mud. So we called a tree guy and for a small amount of money he brought a huge load of chipped wood, and it all went into the run. It has helped quite a bit. We did have to add flat boards around the edge of the chain link fence to keep the chipped wood inside the run. Those girls love to scratch and dust bathe.
Your run looks like chipped wood would help immensely. A roof will also help, but for now, the mud needs to be addressed. Its a process and takes time!
Good Luck!
