I have a small flock and during the day I will let them forage in an outside run. The only predators I have seen are large hawks. I do have coyotes, raccoons, foxes, etc. but I have never seen one near our run (likely because of the three dogs patrolling the yard). My run is only fenced in...
You can cover the top with deer fencing, it's strong but lightweight. You wouldn't need to to build rafter supports, it can be supported by rope, cable, or paracord. A 7' x 100' roll of deer fencing costs about $25 and you would only need half of it to cover your run.