First thing, the motor can't do anything on it's own, there should be no power to it at all until a button is pushed. First you have something closing in one of the switches or the wiring shorting together to feed power to the motor. Relay has to be activated by power from switches to work be a real fluke for one to short or close by itself. Clicking relays don't always mean they are working properly. Usually when your motor runs after tapping on it means the brushes are sticking or armature is dirty from brush dust. Most can be disassembled and clean fairly easy but if it has gotten hot several times it also have a magnet that has come unglued inside motor. But first we need to find where power is coming from.
#1 check under cowl and check all wiring to add on switch, make sure nothing under ther is shorten switch connections, then check the wiring run for anything rubbed together. #2 Then trace wiring back to switch in your hand control. It's going to be a process of elimination. In a situation like this I always try to eliminate any add on stuff like the cowl switch. And any connections where it was spliced in.
#3 once you determine what is causing the problem and repair that, if the motor is bad it can be rebuilt. Jones trolling motor a board sponsor on here could probably rebuild it or might be able to get you a replacement a lot cheaper than factory.
Hope this helps some.
Proud Member of Team Geezer
Charlie Weaver USN/ENC 1965-1979