Cedar planks and mothballs are what I'm aware of that are effective in repelling moths (and the gases from the mothballs will kill them but only if they are in an area enclosed enough that the gases can build up to a high enough level).
One thing that might help prevent the moths from even getting to the table surface would be to get six large sponges of the car wash or similar type sponges and then scrunch each one up sideways and insert it into the pocket in such way that it expands and completely seals it off making it difficult or ideally impossible for moths to reach the playing surface through the pocket opening. They may still be able to get in anywhere the table cover isn't sealed tight enough against the rail, but it cuts off the large pocket entry points that would be real easy for them to find.
Now put cedar shingles, or mothballs, or both, on the table surface. I don't know if mothballs can stain (it is some kind of a chemical) so I would put them on something other than directly on the cloth just in case, and if using both you could just put the mothballs on top of the cedar planks I would think. Not sure why the links below are showing as "robot or human" but they go to a couple of the cedar type products that you can consider but any cedar scraps should do.
The cedar tends to smell good to most people, and mothballs tend to smell kind of bad to most people, and until you know if the smell will permeate and stay in your pool cloth for a long time you may want to take that into consideration when choosing which to use depending on how long your let it sit covered without playing, and depending on how good and bad you personally find their smells to be.
I think putting the sponges to seal up the pockets real good, plus the use of the table cover, will help to hold in the cedar or mothball scent much more than it would have with the open air flow from the pockets and will make the cedar or mothball scent much more effective since it will stay much more concentrated. I could even be that the moths can't get to the table surface at all to begin with when the pockets are sealed and the cover is on, but between the two things (sealed pockets and a moth repelling scent on the table surface) along with your table cover I'm guessing the moths aren't going to mess with any of the table surface or rail cloth, just too tough to get to it, and too concentrated an odor they can't take.