Yes - run a jump lead from the battery positive to the main starter motor feed in addition to the one already there, and run the other jump lead from somewhere on the engine.gearbox to the battery earth. This will double the wire thickness to and from the starter, so if it still struggles, it is definately the starter motor at fault. If it now works OK, then you'll need bigger wiring to the starter than is currently fitted, or will need to improve their connections.

As your mate said, either the starter motor is drawing too much current, or the wire can't pass enough current, causing it to overheat.