ya with the cai and the fact that you said it has to be warmed up makes it seem most likely like fuel cut, if it gets worse when it's colder out like at night (depending on where you live) then that is even more evidence that it is fuel cut, the tune on these cars are just really bad, I really suggest saving up for an ssafc, look on Protegegarage.com for it, it is worth the money big time, and don't get the dealership reflash, just tune the car, because once you reflash it, as far as I know you might not be able to go back, but really even if you can, your better off just tuning it correctly with the ssafc because all the reflash does it retard the timing instead of correcting the fuel maps
I think with these cars, since the stock air box and everything is in the engine bay, and with the stock hood, the engine bay area is really really hot, so it is sucking in really hot air, the car is tuned for that hot air, so with a cai, the difference in air temp and air being read from the MAF sensor I think throws it off enough to cause fuel cut when it's not really hot outside.. at least that is what I think