Yes! Awning above coming off the coop roof! Kudos! Make it wider than the nest roof and maybe 1/4 or 1/2 as far outward. I would still slant the nest roof.
I think that would help. However, you still have the problem of rain hitting the sidewall and flowing down on top of your flat nest box top.
If you have to redesign your nest box and rebuild part of it, I would think slanting a fixed nest box top and having the end drop down would be easier than modifying your roof. Also, ascetically, I don't know if an awning on the roof above the nest box would look all that good. More importantly, you still have a flat nest box top that will still pose problems.
Better just to redesign your nest box like the pictures provided by @Club cramahe coop. But maybe if @BielefelderHen (OP) uploaded a picture of their setup other options might come to mind.