I don't know how wide a span you are trying to cover. The wider it is, the more load on it. Snow and especially ice can load up a wire or fencing roof as well as a solid roof, so you have to figure snow and ice load for wire too.
If you put a flat roof on there, it will leak. I don't know of any feasible way to make a flat roof that won't leak unless you can use a solid sheet of some material. I like a solid roof for shade even if it leaks, but the snow and ice load can get pretty high if yoou have much of a span. You need decent support, which gets expensive.
Foxes and raccoons can both climb, so yes you need a top on it to stop them. No matter how predator-proof you think the run is, I still recommend locking them in the coop at night. It can be done, but it is harder to make a run truly predator proof than a coop, especially a run of any size.
On a budget, I'd give real strong consideration to using 2" x 4" welded wire up there, maybe 14 gauge. Snakes, rats, some members of the weasel family can get through that, but it will stop foxes, raccoons, and other large predators except maybe a bear. It's real hard to keep snakes, rats and some weasels out anyway, so depend on your coop for them. I'd use wire to attach it. Not plastic zip ties or anything plastic because that will deteriorate over time, and not that long a time either. You could cut lengths and attach the wire at single points to the top of the kennel, but I would use a long length of wire and wrap it continuously in a spiral.
You will probably have to splice strips of the wire together to get a continuous cover. The fencing probably won't come in a wide enough section to cover it all. You can use wire, hog rings, or C-clips to splice the wire fencing. If you have the spans I think you do, this top should hold up to snow and ice and be about as inexpensive as anything I can come up with.
Good luck!
