This white wire feeds a switch loop that provides power to the switch for the fan. Unfortunately, there are a coupleseveral problems here:
- This white wire needs to be tagged with black tape to mark it as a hot
- It's not clear where the neutral comes from, but it's clearly not coming from the same box as the hot. This is at minimum a 300.3(B)/310.10(H) violation due to the wayward neutral (and a great way to screw up things sensitive to magnetic fields), but may also be a much more egregrious violation due to someone abusing the ground as a neutral.
- There's another junction box somewhere in this scheme that's doing something odd. You'll want to find it before doing further fiddling with the circuit.