I think I'm your man on this one, I do a lot of zoos and shooting "through the fence" is my motto!
The photograph posted isn't a very good quality one, so I can't exactly comment on it, I think uploading through TPF is reducing the IQ. I'd just tell you what goes through my mind when I'm out shooting.
1. Is the subject too close to the fence? If yes, forget about it and move on, or wait for it to move back.
The ideal distance is at at least a three feet away from the fence for the lens to focus on the subject without the cage halos, and at most whatever your lens can reach.
2. Is the light coming through the fence on to the subject, or from up above? If former, try to avoid the cage shadows on the subject, getting rid of it in post is very difficult.
3. Is there a fence in the background too? If so you'll have to work extra hard to try and find a better background, or your shot is ruined.
4. If all the parameters are satisfied, I touch the mesh gently with the lens and put my weight in it slightly to avoid vibrations, and wait for the flimsy mesh to stop moving. I always switch to manual focus, and take a shot keeping all the above points in mind.
Even after following all this, I sometimes get mesh halos in some shots. I usually discard them, but if it's an absolute keeper, I've noticed that reducing the black point and increasing the contrast in Lightroom helps with removing the halo a lot and the rest I tweak in photoshop . I'll post an example for you, you can notice the halos in the before:
Before:
After: