If there is nothing on the top of the run, you have the risk of chickens getting out, as well as predators - dogs, possums, coons, hawks, owls, etc - getting in.
With the run being only 4 feet tall - having it covered is going to suck and kill your back when you have to get inside to catch chickens, clean, refill water/food, etc. And it is not fun to have to catch a chicken and wind up getting covered in chicken poop because you had to crawl on your hands and knees to move around in a short run. So think about raising the support posts to at least 6 feet tall if you decide to cover the run.
As far as chicken wire vs hardware cloth - that depends on how much you want to spend as well as how the run will be used etc.
A predator can reach inside chicken wire and pull a chicken's body parts off. And small chicks can still get their heads through the holes of chicken wire - so if you're going to have small chicks next to a dog pen - hardware cloth would be better at chick height. If your chickens are going to be put up at night and locked in a coop, then putting up chicken wire in the run may be fine, and then you can line the windows of the coop with hardware cloth for predator protection.
We have very large chicken tractors that we drag around the pasture. The runs are wire fencing with holes that are about 2 inch x 3 inches. The tops of the runs are covered with this wire also and shade cloth is placed on top of that with tarps on top during winter. Then we run 3 foot tall chicken wire along the bottom, inside the welded fence wire - that's to keep chicken parts inside. The attached coops can be locked and the vents and windows of the coop are covered with hardware cloth. Because we have ventilation under the roof, there is hardware cloth covering the roof, underneath the roofing panels, to keep a climbing predator from trying to get in under the eaves of the coop. We also have some open air pens that also have the welded wire/chicken wire combo on them and plywood shelter on one end that is 4 ft deep so a predator isn't easily able to reach in and grab a chicken that is in the "apartment" area.
It's all about what your predator threat is, what your budget is, and what your coop/run design is and how it is working for you.
I'm set on 6 ft hardware cloth sections with the hardware cloth on top after reading your reply. Now just have to get the hubby busy! Thanks!
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