Dear Ernemats,
That is the bracket for the second starter solenoid. You should have a 1986(ish) Ford starter solenoid on that bracket. (See image below. The exact year is on a different post.) The lead from the ignition key goes to the second solenoid, which switches power on to the actual starter, which is pretty finicky about having enough voltage. When it gets hot, the rather circuitous routing of the starter solenoid wire can leave it at under 10V, with the problem that you are experiencing. With a little cooling off, the battery recovers, and the voltage is high enough to start the engine.
Of course, you can have the problem that even with the second solenoid, the voltage is too low, but that's another post. (I think Charlie Iliff and I are going through that at the moment. I think that the ignition switch may have too high a resistance in it.)
All the best,
Peter