One method I've heard that helps stop this is to have your fence about 6" higher than your posts, not anchored to anything. This would have the effect of not providing anything solid for the birds to land on, as they would most likely fly to the top, perch, and then fly down.
Before I covered mine, I saw one of mine (a BR)attempt to fly to the top of my wire fence. She got to the top (6' but she launched from a 3' high roost), but immediately fell back into the run when there was nothing but wire for her to grip onto.
For an inexpensive cover for your run, go to the garden section of your home improvement store and buy some bird netting It's black plastic and usually comes in a roll that's about 14' by 14'. It's lightweight, but the birds recognize that it's there and don't challenge it. I ran baling wire back and forth across the run to provide some support, then attached the bird netting to my fence with zip ties.