My PSU and motherboard are brand new, but the 24 pin plug from the PSU is really loose in the motherboard. Like if I nudge it it falls off.

There is a pin missing on the plug from the PSU that is not missing on the motherboard so could that be a problem? This is frustrating.

The power supply connector is supposed to have a hook which latches onto the connector on the motherboard connection. Or are you saying the PSU is modular and the connection from it loose?

There shouldn't be any missing pins on the plug, but even a missing pin wouldn't make it loose, everyone one I've ever put in has been extremely tight, especially when new.

[Insert reference to boning here]