First, does the water gather/puddle there in the run, was there standing water there before chickens, or drain off to adjacent areas?Has anyone else dealt with this kind of thing before?
Gutters on nearby roofs and/or trenching and/or swales can keep some of the rain water out of the run.
How about a real roof on the run instead of tarps?
You probably need more organic matter...a good mix of sizes, shapes, materials to 'eat' the poops and ameliorate the mud. Straw can be an ingredient, but bigger wood chips and other items are needed to balance the mix.maybe because it's mixed in with a bit of straw and organic matter
Here's a great description of contents and how to manage organic 'bedding' in a run or coop...and there's a great video of what it looks like.
http://www.backyardchickens.com/t/1037998/muddy-run-help-please#post_16017992