Here’s a winch wiring diagram–it’s basically a larger directional starter motor.
Two basic circuits–the controller circuit and the motor circuit, connected by a contactor (heavy duty relay). There’s additional provision for a remote controller (the socket you have) and a circuit cutout switch. Basic winches only have a dash-mounted controller.
Large winch motors draw a lot of amps, like a starter. So they use heavy 8 gauge wires; not many switches will stand up to 40 amps, so the controller is wired up to a 10A power source which energizes the contactor, closing the motor circuit. You can wire the controller circuit to the ignition circuit in several ways–the importance being that the ignition circuit serves as the cutoff. Because the motor is reversible, the controllers have three wires: ground, winch in and winch out.
That accessory plug you have there can be used to power the controller circuit–it’s disabled when ignition is off. This circuit is best used for something else IMO like a phone/GPS charger. Yours is already wired up you just need to figure out how to find/replace you controller(s). It looks to be an older warn remote
Here’s a last pic—a cautionary tale. Winch controller circuit wired directly to battery and wires were exposed on the controller. They shorted out and I couldn’t stop the winch so I had to tear off the controller and separate the wires.
