John, installing a gravel bed in combination with a submersible pump will fix the problem for well under $100. Once you have the necessary tools, equipment, and supplies, you can have the fix completed in two or three hours. Here's what you need:
Tools:
Shovel
Utility Knife
Channel Lock Pliers
Drill w/ 3/8" bit
Equipment & Supplies:
Submersible Pump w/ vertical float switch
Discharge Hose
5-gallon bucket w/ lid
Coarse Gravel (Lime is a good choice.)
Procedure:
1) Just outside the lowest end of your coop area that is covered in gravel, dig a hole about 3' in diameter and a couple of feet deep. (If necessary, dig a trench from the coop gravel field to the hole.)
2) Place a few inches of gravel in the bottom of the hole.
3) Drill 3/8" holes in the 5-gallon bucket: Starting at the mid-height of the bucket, drill 8-16 holes (the more, the better) around the perimeter. Make about four rows of holes working toward the bottom of the bucket. Then turn the bucket over, and in a circular pattern, drill holes in the bottom. (These holes are important because they will allow water from the gravel field to drain into the bucket and be pumped away.)
4) Place two or three inches of gravel into the bucket, and the place the bucket into the hole.
5) Fill the hole (and trench) with gravel to ground level, but do not allow any additional gravel to go into the bucket. (The top of the bucket should be at least a couple of inches above ground level; however, the bottom of the bucket should be below the lowest point of the coop drainage field.)
6) Place the sump pump into the bucket on top of the 2-3" layer of gravel. (If you don't already have a pump on hand, I recommend the Harbor Freight 1/3 hp Submersible Sump Pump with Vertical Float 3400GPH [Item#63321]. With 25' of vertical lift, it will give you more than enough power to overcome any hill or slope. The pump is $70, but you can get 25% off if you buy it on the 4th [
https://www.harborfreight.com/]. [You can also get a discharge hose at HF; however, they may have an adapter that will allow you to use an inexpensive garden hose as a discharge hose.])
7) Use the utility knife (or a hole saw) to cut a 2" hole on one side of the bucket lid. (This hole will be for the discharge hose.) Then cut another 2" hole on the other side of the lid. (This hole will be for both the power cord and float-switch cord.) See page 4 of the HF pump user manual:
https://manuals.harborfreight.com/manuals/63000-63999/63321.pdf
8) Thread one end of the discharge hose through one hole in the lid, and attach the discharge hose to the pump; thread the two power cords from the pump through the other hole in the lid; and, then snap on the lid onto the bucket.
9) Place the open end of the discharge hose to the area where you want to runoff water to drain.
10) Plug in the power cord and the float-switch cord. Note: If you have standing water in the bucket, the pump should immediately engage and begin draining your coop area. The pump will remain on until there is about 1/4" of water in the bucket. Once it reaches that level, it will automatically cut off. (If there is no standing water in the bucket, you can test it by filling the gravel bed that surrounds the bucket with water from a garden hose.)
The above sump-pump and drain-field set up should keep your coops high and dry, and the gravel bed will keep trash from getting in to the pump and damaging the impellers.
Fortunately, my property is at the top of a slight slope, so I don't have any drainage issues. Everything just naturally just runs off. However, if I did have standing water, I would definitely put in a sump pump. They're relatively cheap, and they're easy to install. Whatever you decide to do, best of luck!