Im rusty as an old nail on the latest Code, but best I recall you can use direct burial rated cable like (UF Undergroud Feeder) or if you wanna bury conduit you can use individual enclosed THWN rated wires. The W in THWN versus say THHN wire is for wet locations as I recall. Of course just trenching in UF cable is easier but Id go deeper then only a foot, warning tape above the cable isnt a bad idea. Plastic conduit is permissible unless under driveways and it needs to be rigid or IMC when it comes up above the ground and enters a building..........
If shes tripping a breaker even when opened and capped of at the pumps end, must be a bad/shorted/chewed cable well DUH
Have you calculated voltage drop at your load?? If necessary you may need to increase wire size??
John T No warranty Im rusty remember