It's nothing to do with the alternator. The basic starting circuit is the battery, the starter solenoid, the starter motor and the wires connecting them. The starter solenoid is essentially a relay that connects the power to the starter motor and is mounted directly on the starter motor. It uses a smal current from ignition switch circuit to connect a high current path from battery to starter motor.
The circuit is high current so it uses thick wires. These are the positive from battery to solenoid, the earth between the engine/transmission and the bodywork and the earth between the bodywork and the battery negative. There is also a short wire between the solenoid and the motor but it's part of the motor/solenoid unit. Earth wires can look like regular insulated wires or just thick braided wire.
The clicking noise is probably the solenoid trying to energise then dropping out again because the voltage has dropped. The voltage has dropped because there is either a poor connection or the battery is suspect. I wouldn't suspect the battery if it starts OK when cold (that's worst case scenario for a battery). So, poor connection somewhere, obviously getting worse when warm. It's a case of checking all joints in the circuit I described above. I would disconnect/clean/tighten at both battery terminals and at each end of any thick earth strap/cable.
The tricky one to access is the terminals on the solenoid/starter. It's at the rear of the engine and probably best reached from underneath (with the front wheels on ramps). Check everything else before suspecting this.
Failing all else, it should be simple to find the problem using a voltmeter if you have one. Post again if you need more guidance on checking voltages.
Red
