The 24-pin should clip on, so if that is actually fixing something I would look at the 24-pin plug and see if any of the wires are trying to back out of it. It also wouldn't hurt to tripple check all other connections are solid...
It's quite random and when I wiggle it around I don't think the connector moves too much it is clipped on very solid, almost impossible to take off but that could be because it is @a 90 degree angle (EVGA Z68 FTW)
I can try to unplug it and see if that helps (wish me luck)
Any chance the wiggle effect is reaching as far as the PSU itself?
Anything look unusual up at that end?
Well taking a closer look it looks like one end of it was a little out which is weird because the clip is perfectly tight, this might fix it but wiggling it shouldn't reach to the PSU because it's going through two grommets in my case (800D)