Wood chips asap. Call a tree service in the morning. After you get them down you will be amazed. Now you need to add other thing to it like leaf's, yard clippings, dump the grass catcher, dirt, pine shaving from the coop... But just the chips get you out of the mud. Most tree service will dump for free. No more than $35 a truck load. Make some calls.
Deep litter can help stabilize a muddy run. However if you have existing drainage issues (hard clay soil, run is in a low lying area, you live in a "dry" lake bed) then drainage needs to be addressed first.