You found a small voltage drop in that section. Very much agree the heat is a big clue. 5 amps in that wire should not heat it any more than the other 10 and 12 gage wires.
I'd start looking for bad connections and green oxidized wire from the relay to the ammeter.
Definately remove the connectors and pop out the metal terminals. Its not that hard. Take your time.
Same thing with the blue run wire connector and terminals. The drop you found there is even worse. Problem(s) also could be at the start switch, or worst case, the main splice. So check the whole run.
Does that make sense?
