What do you do with this problem?
I saw a wasp circling around our outdoor area, I am afraid of them, so I removed the birds, closed the house door (our outdoor area is on my front porch). Then I saw the wasp actually entering our outdoor area thru the mesh. Do you have any large flying insects where you live? Are these insects harmful to the birds? Anything I can do to keep them away?

Can you add a layer of window screen mesh over the outside of your aviary mesh?

probably not. at least not easily

Stinging insects are potentially dangerous to birds. Mosquitoes might be an even bigger danger than wasps since they can carry West Nile virus, and they actively seek out warm-blooded animals to feed on. I don't know a solution though. Insects aren't much of a problem where I live.