It's a pretty safe bet that whatever you have existing doesn't meet code. There's a fair chance the plug is cut off because there's a short in the line. You'd need to use a cable approved for direct burial like UF-B.

It sounds like somebody hacked it up with inappropriate and unapproved wire, and the wire failed. I think I see some digging in your future.

The only way that could not be a code violation -- and not be potentially dangerous -- is: 1) if the light is low voltage (12V), and the wiring is, too; or 2) that wire is part of the controller for a photocell and the power comes from elsewhere. Can you tell by inspection if that is a regular fixture or low voltage, or if you can see a photocell anywhere?

If you use rigid metal or intermediate conduit you only need to bury the conduit 6 inches deep.