My friend, BigJim, may be onto something about over-gripping. In Skeet, the follow thru rule is, "stay on the target until you see it break" -- one of the pitfalls of shooting with a pump is "shucking the cob before the water's boiling." If you've never lived on a farm with fresh corn, it means you're concentrating on the gun instead of the target. It's one of the reasons I recommend against new shooters using a pump. With doubles and a pump, many newer shooters wanna pull back on the foregrip as soon as they touch off the trigger. Consequently, their follow thru and rhythm suffer, and they start doing the same mistakes in single presentations.
IMHO, cycling the action should not be part of shooting singles. Shoot, follow thru until you see the target break, only then do you work the action. With doubles, shoot the first target, follow thru until you see it break, reverse the gun and then cycle the action. Working the pump should be the beginning action on the second bird, not the last action of the first.
I know I'm splitting hairs, but in singles and the first target in doubles, the gun action should be the same no matter what kind of gun you're shooting. What Big Jim and I are saying is, you don't want a dead gun while you cycle it. If you cycle before you move to your second target, then there's a very good chance you'll stop too soon and jeopardize the first target. You can't get a dead pair if you don't break the first one. After you practice for a while, you'll realize that you really do have plenty of time to cycle the action while you're bringing the gun to the second target. Another consequence of practicing your doubles this way, your singles will be much smoother -- your subconscious won't be worrying about getting a forearm death grip so you can instantly shuck when you shoot.