Welcome to the forum, sloveladypowell. As you can see,
, this forum is against outdoor housing for various reasons - animal predators, hot and cold weather, and of course outside, you would most likely not interact with them as much, therefore maybe not noticing illness or change in behavior. But I think you need make the best of your situation at the moment (making sure the cat cannot get to them)
I have no experience with allergies, but maybe you should get tested again, as it may be hay or their fur which irritates you. Using fleece bedding or serving their hay in a paper bag might help.
As they are currently outside you do need to protect them as much as you can. I would recommenced constructing a solid house for them to stay in at night, and shelter from wind, rain or hot sun. Make sure the floor is lined with plastic to stop urine spreading into wood and becoming stinky. Then i would make sure the run has a roof that can't easily be blown or knocked off, maybe chicken wire? If it's too large to cover at the moment I would solidly cover some of it and let them sue the rest while you are around to supervise. I'm not sure of the weather conditions where you are but you will need to adjust their bedding to make sure they are comfortable. If it is very cold/hot try and bring them in overnight. Or - better - do you have a shed or *unused* (no poisonous fumes!) garage you could house them in?
overall I would try and arrange for them to be inside...maybe a little-used room or something? Anyway as they are outside and apparently liable to attack I thought I would try and help them stay safer outside.