Shade tolerant cover/forage crop for run

[aart]

You could try grazing frames...will keep the chickens from killing the plants.
https://www.backyardchickens.com/articles/grazing-frames.73645/

 

[paintedChix]

Here are some other ideas of grazing frames... I'm still looking for a specific video.

Here's another one -

Found it - at about 3:49 -

another video shows where he's put those fence rings around a small compost area, puts in sprouted oat seed (even in the winter) and then when it reaches a height he removes the ring and allows the chickens to "graze" that section of "pasture".

and there is this one - since he doesn't "grow grass" in the actual chicken run which is used for his composting -

He has many videos on chickens, chicken runs and compost. His chicken run isn't the most sunny, so you may be able to use some of what he does. Over the years, since his first video when that same chicken run was "naked" (bare, bland looking soil), it has made an amazing change...

To make the area a little less "shady", you can have the trees "topped" and/or pruned by a service. Then compost what is trimmed off.

You can plant perennial shrubs, that bear fruit in acid loving soil and some shade - blueberries,silver berry, elderberry and others. You can plant shade loving herbs that will also work for the chickens.