We have a Large park that is lit at night by a underground 3 phase power. That lights come on via a relay that energizes at night. When I take a voltage reading of power coming into the relay, each phase to ground is 277 and phase to phase is 480Vac. But when I energize the relay turning on the lights, two of the phases to ground read 480Vac and the other only reads 3.8Vac. What's causing this? Short to ground?

I will assume the main is 277/480 volt with a grounded neutral, and the lights are 480 volts single phase connected in Delta on the 3 phase conductors.

If it was a short to ground, I would expect a fuse/CB to open.

What is the OCPD? Fuses or 3 pole CB?

I need to suggest that even though you see 277 phase to ground and 480 volts phase to phase, that you start at the beginning of the 480 volt system, and confirm no fuses or breakers are open, starting at the Main, and work your way through the system to this lighting contactor (relay).

I would then disconnect the load wires from the contactor to confirm proper voltage at the load terminals of the contactor with no load.

The odd voltages you get phase to ground on two phases and one goes to 3 volts leads me to believe the supply feeder is the issue, rather than an issue in the lighting circuit.

This is a start to troubleshooting.