One thing that you can do is make sure that you have a decent kite. Any sign of a tight leech and gust response goes out of the window.
Our sails are a couple of years old, but still in ok (though not great) condition. Would having the luff a little too loose have a serious effect upon gust response? The tack line we're using has a bit of stretch to it (probably 1-2 in. when we heat up in strong winds) and I've been meaning to replace it, but haven't got around to it yet.
Thoughts on spin trim in response to the puffs? Are you sheeting out or in, as(before?) the puff hits?
When anticipating the puffs, (particularly the ones I'm describing which were accompanied by a header) I'd say we were sheeting out the spin a little and bearing off before the puffs hit. Once the puff hits we're heating up enough to keep the windward hull out of the water.
In the case of when we didn't anticipate the puff at all, probably we're sheeting out a little and *trying* to bear off as the puff hits, but mostly we're just going over.