Two quick solutions. It looks like you have a good overhang above the coop door, and that wind blown rain was not the likely source of your water intrusion, so...
1.) Increase the height of the sill at the base of your coop door. (same principle as sand bagging around your home during a flood - the higher the sill, the higher the water must be before it enters.
2.) Looks like the water is being partially directed by terrain, coming from the run towards the house. A shallow ditch or the like will help direct it towards a different location. this will also reduce static pressure, so you don't find water being forced up thru the ground when you have standing water outside.
So, a couple minutes to a half hour with a shovel, maybe a few minutes with a hammer and a couple nails, and you should have a low maintenance solution for the ages.
/edit you could french drain, too, but as the above poster said, that's a lot of work. I created a bowl to direct water towards inside my run, away from the house, then ditched it to a low point away from run and hen house. Not as attractive, perhaps, but certainly effective.
Outside the run, a combination of berms and ditches built over the last few months helps keep an acre worth of higher elevation property from draining through my run, or staying there afterwards.