Have you tested it with a multimeter?
I'm not an electrician or similar and other than installing a relay for my watermaker
I have no experience with them, but I have a slight clue about how to trouble shoot problems like this.
The heat you feel is possibly caused by some high resistance caused by a fault in the relay and this is how I would go about tracking down the problem.
Set it on the Ohmmeter setting to test the resistance. If you are not familiar with it's function then this is what you should see.. When the terminals on the ohmmeter are not connected, then you will get no reading. If you hold the terminals together, you should be making a good circuit and get a low resistance reading, very close to zero. if you hold the terminals in your fingers, you will make a circuit with your body, so you will get a reading which shows the level of resistance in your body.
With all wires disconnected from the relay, use the ohmmeter on terminals 3 & 4. There probably should be no circuit made and therefore you should not be getting a reading. If you do get a reading, then the circuit is always on and that's a problem.
Then test terminals 1 & 2. I'm not familiar with relays, but I would imagine that you probably should be making a circuit and therefore get a low, close to zero reading. If you get a higher reading (like when you tested it with your body) then this could be the resistance that creates the heat.
Finally, connect the wires back on to terminals 1 & 2 and assuming its controlled by the ignition turn it on. If you are not sure, test using the volt meter to see what sends voltage to the wires on 1 & 2 (should be + & - wires). When there is power to 1&2 then test terminals 3 & 4 for resistance again. If the relay is working, then you should get a low (close to zero) reading.
In all likelihood there is a problem with the relay and they are dirt cheap
from an automotive supply place, so buy one and then do the tests on both to make yourself familiar with them. I wish I had one handy now as I'm curious to test to see if I am talking complete BS or not