I just got back in - it is cold out there!
Here is my guess at what is going on and what to do.
I think the French drain is the key to this. It is supposed to carry water from the East (down the steep slope into the French drain and away) and you can see from the water flow that it is also pitched to take water from the North West (where you drew the arrows of visible water flow).
If it is stops working, or is simply inadequate for the volume of water then you will get pooling at your lowest point which should be next to the French drain - and clearly is because that is where you get pooling water in the chicken coop.
French drains without landscape fabric get clogged with soil and even with fabric you sometimes have to flush them out.
So, I think the first thing is to really establish if that French drain is working - should be easy to test by seeing if water emerges out of both ends (I am still trying to fathom why a U-shaped French drain is needed but that isn't important for the chickens!).
If it isn't working it is worth trying to clear it. The problem is that it may be the stones that are bunged up rather than the pipe itself. I don't have any good ideas for how to do that beyond a pressure washer and some digging around with a pickaxe.
What I fear is that the French drain actually needs replacing. That is a big job and not one I could do without a pro to help.
In case you or hubby go - 'all that for a chicken coop' - my thought is that a failed French drain is going to get you into a swamp and probably compromise the barn it was built to protect - so it might need to be done anyway.
I would bet good money that solving the French drain issue will solve the chicken coop issue, but just in case I would probably do something on the North side.
On my own I would construct a berm of compacted dirt to divert water away from the coop and into the French drain - could also be a low wall or even some of that plastic garden edging. Anything to make it easier for the water to head East rather than SE.
However, if you are going to have to get someone in to redo the French drain I would probably ask him to throw in a mini French drain along the North face of the coop.
I think the damp areas in the South part of the coop are wicking from the drainage issue which in the North and so I don't think you need to do anything specifically for those.
Short term, I would throw a pallet or something like that over the spot marked 'very wet' until you can sort out the drainage.
I know this is a major pain but I am sure it will be worth it in the end!
Drainage tax: Sylvie, another judgmental chicken!
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