I would try 1 of 2 items. Maybe a tarp, made of thicker plastic grade..pool cover type for the area that isn't covered with screening. There should be enough ventilation for the pups when outdoors with what you already have. Check Craig's list maybe for a used, smaller pool tarp that can be cut..

Also, we just purchased screens for 2 doors. First is for a large French door that secures with Velcro top, bottom and sides using whicher ones are needed. It can be unhooked at the bottom to let the dogs out on our patio from the great room and also into the fenced backyard. I have a smaller size on the kitchen bay-canto-levered door/window. So there are muliple and various heights X widths. Another door store offered custom orders for any size.

Home Depot and Lowe's carry them in different sizes and its very durable, mobile and thick, more so then regular screening..

I would buy the bigger cattle panels and lay over the top..they are like 4x4 inch squares and are like 4x 16 or 20 ft. and put them over the pen....or there is a really light weight wire...welded wire that is really inexpensive that you could put over it. But actually anything even like netting would detour hawks or owls.

Do I have to tell my Hawk story again ? [Ch.Shalane Fly By Knight , call name Hawk , there is a reason for that ]

I string fishing line over the puppy pen. I cross cross it and then tie heavy yarn or pieces of rag to it here and there. The hawks don't come near.

We have had a Hawk attack. When we rebuilt this year I had to knock myself up the side of the head for not ever thinking of this. We had a litter moving in and had to do something fast!

I took some cheap bird netting for the fruit trees and stretched it across and just tacked it here and there with plastic ties for a quick fix. It is still up and keeping the little birds out too. It was the easiest and best thing we have ever used.