There are two approaches to solving a water problem. First, try to keep the water out to start with. That may mean covering it to keep rain out or diverting water so it doesn't flow inside. The other is to get the water out.
Not really. It's kind of a lower area so water seems to collect. If I can build up the soil it should remedy the situation.
This is the problem. It is a low area so water drains to it. The soil is probably an impervious material like clay so the water stands instead of soaking into the soil.
When I built my coop I built it at the bottom of a slope but in a flat area. I put a berm and swale above it to create a channel for water to flow away without getting to the coop and also put 3 or 4 inches of that soil in the bottom of my coop to make sure it was above the water level even in a heavy rain. That coop stayed very dry. I don't know if your terrain is suitable to a berm and swale system.
French Drain can mean different things to different people. The basic idea is to give the water a channel to flow to a lower spot. Basically dig a trench to a low spot and fill it with a pipe, gravel, sand, something water can travel through. Don't block the outlet so water can escape. Don't block the inlet either. If the soil is an impervious material like clay water cannot get to the drain. For this to work you have to have a low spot for it to drain to.
I have not done this to my coop or run. I had a gravel driveway that was blocking the flow of rainwater runoff so it was backing up in a low part of my yard and causing a part of that driveway to be soft and mushy. I dug a trench through my driveway and filled it with pea gravel. On the upstream side I dumped a few bags of pea gravel so the drain could collect water. That solved the problem. The driveway was no longer mushy and most of that water drained away fairly quickly after it stopped raining.
There are different solutions depending on your terrain, soil types, and other conditions. Good luck on finding yours.