I have similar problem, will check my relay, here is what I just left in a new thread
Sometimes the mower will crank-over with the key no problem and will start, however, if it doesn't start after cranking-over the first time, it will not try to crank anymore using the key until maybe the next day. I can leave the key on and jump the aux. solenoid and the mower starts no problem. Something is changing after the first engagement, which causes the electric not to reach the aux. solenoid/starter.
Originally this mower had a wire going directly from the key switch to the starter/solenoid, however, I added an auxiliary solenoid in order to spare the Newly installed key switch from excess electrical heat from the direct starter/solenoid hook-up. I don't know of any reason this set-up should not work the same, (I believe the newer models actually addressed the wires overheating by including this set-up) The mower will crank with the key, normally on the first attempt to crank, then it will not crank with the key again until the next day or so.
Is anyone familiar with the kinds of electrical gadgets on these mowers which might do this?