I recently added two cyclone fence kennels to my run for added space. I tied down 10 foot metal poles across the top, then covered it with hardware cloth and secured the cloth with hog clips. The weather will go through the cloth.
DO NOT USE A TARP...I did that on my first run and it was a pain. The rain collected, the snow collected and we had to lift the tarp everytime we went out there to get the water off of it and it made the tarp hold water and swell down into the coop.
I also suggest you cover your chicken wire with hardware cloth...up the sides from the ground three feet to keep predators out. Chicken wire holds chickens in, not predators from getting in. Great looking coop and run!!!!!
I strongly suggest whichever material you cover your run with, that you slant it so that heavy snow can slide off. I live in Oklahoma where we typically don't get much snow, and my run is covered by chain link fence panels. Last winter I put a tarp over part of it to help keep part of the run dry. We got much more snow than usual and it gathered on top of the tarp so much that it caved in the top of the run.
You've got the perfect setup to extend the roofline of your permanent structure and create a "carport" kind of effect. You'd have to tear down most of what you have now as far as the posts and chicken wire across the top. But, may I say that doing this would not only look great and protect the run from predators, snow, heat, etc. but would also probably increase your property value since such a structure could be repurposed for various future uses...like a carport.