I wonder if it might be too shady there, shaded by both the house and the tree. Does it, perhaps, get direct sun in the morning? That would be ideal, I think.
Personally, I don't think I would cover it with anything just yet. (Is that 6' chain link, or is it shorter? It's hard for me to tell.) Have you actually seen any kind of wildlife inside
the run yet?
If the run gets no more than 6 hours of direct full sun every day, it might be better just to roof it over with something like plastic deer fencing. That should keep out everything except insects and smaller rodents. Unless your area has a problem with roof rats, as exists in some parts of Florida, I think even the smaller rodents would go for easier fare, before bothering your veggies. A determined rodent would be more likely just to dig under the fence, anyway, rather than climbing it and gnawing through whatever you use as a cover. (Is the chain-link set in or on cement footings? That's how I had mine built, so I'm just curious.)
If the run gets less than 6 hours of unobstructed sun every day, I think you might want to consider moving it to a more open location. Most plants that produce fruit, like tomatoes and cucumbers, need a minimum of 6 hours of full sun each day. In your area, morning sun would be kindest to the plants when the temperature is high.
You could cover the greenhouse with heavy plastic in the fall, removing it in the spring, to extend your growing season and, perhaps, grow cool-season veggies during the winter.
(I garden in my dog run, too, btw!
"Isn't it enough to see that a garden is beautiful without having to believe that there are fairies at the bottom of it too?" - Douglas Adams