How big is your run and how tall is the fence? The bigger they are the harder they are to cover and still let you walk in there. It can get pretty expensive too. If you are going to cover it, the size might have an effect on what you use.
It’s really difficult to talk about what is best. You get some protection from bird or deer netting. You get more protection from chicken wire the heavier the gauge the more protection. The openings also come in different dimensions. People like to think that hardware cloth is the ultimate, but this link shows you what a large animal can do to hardware cloth. I personally think 2x4 welded wire gives me about as much protection as I can reasonably afford but some smaller critters like snakes and many weasels can get through the openings.
https://www.backyardchickens.com/t/1105056/dog-vs-hardware-cloth#post_17001519
Another problem with predators is that you just don’t know what they are going to do. When I was growing up we had two predator attacks the entire time, a fox and a dog. We had skunks, weasels, possum, raccoon, hawks, owls, dogs, and foxes around and the chickens totally free ranged but we only had those two attacks in 18 years before I left home. I didn’t hear of any attacks after that but maybe I wouldn’t. My younger brother did shoot a couple of dogs when a pack of feral dogs trapped some cows and calves in a fence corner but those dogs did not bother the chickens for some reason. After he shot those two the rest of that pack didn’t bother us again anyway.
When I first got chickens here I let them free range (no fences) for three years. I did lock them in the coop at night. I lost two chickens, well-spaced out. I think fox but I’m not sure. I can hear three different packs of coyotes most nights, one to the northeast and one to the southeast on the mountain and one in a creek bed to the southwest. As far as I know they have never been a problem. Then one summer I had two dog attacks that pretty much wiped me out so I had to start over. These were dogs someone dropped off in the country. I really hate shooting dogs and wish the people that dropped them off had been around to see what they forced me to do. After that second attack I got electric netting and have not had any losses to ground based predators yet. That was about 6 years ago. I did lose one a few years back to an owl that went into the coop one night when I was late locking them up and recently lost another, to a bird of prey, probably a hawk. I see a lot of pretty gig red-tailed hawks here all the time and often disturb a pretty large owl when I go out at night but those are the only two I’ve lost since I got that electric netting.
Electric netting is not an install it and forget it solution to ground based predators. It doesn‘t work in snow. Grass and weeds grow up in it and short it out during growing season so you have to take care of that problem. Wind or running water if you get a heavy rain can wash or blow leaves and dead grass against the netting, shorting it out. It does require maintenance but when used correctly it is very effective.
What normally happens is that the predator doesn’t just try to run through or jump the netting, it inspects it first, usually with its tongue or nose. It gets zapped and doesn’t come back. So even when it does not work in the snow it is still a deterrent to those that have been zapped. But new predators are being born all the time, predators that have not been zapped. It is not a perfect barrier but it is a really good deterrent.
Back to your actual question about birds of prey, which you have experienced and which electric netting obviously does not stop. Depending on size of the run if it is very big I’d probably start with bird or deer netting. It is relatively inexpensive and lightweight so pretty easy to support. It will snag on about anything so it can be awkward or aggravating to work with. A wet snow or ice storm can bring it down but that’s true for wire tops too. Snow and ice can get really heavy. If it doesn’t break you netting or wire, it can break your supports.
Is it possible that a bird of prey can zip right through the netting? Yes, but it is not all that likely. Birds of prey have hollow bones which helps make them light enough so they can fly. If they break an important bone they will probably starve to death if something doesn’t catch them and eat them. They will probably avoid the netting for their own safety. That doesn’t mean they absolutely will stay away but it is a pretty good deterrent. It would be my first effort with my run which is not now covered.
