Check the braided woven wire from the solenoid (smaller cylinder bit) to the starter motor (bigger cylinder) for any damage/break or really heavy corrosion.
One click usually means the solenoid is working, the click sound is it throwing the gear forward into the flywheel, the next step which seems to be the problem is it then passes power to the motor (through that brraided wire). Either that or the motor is toast.
A series of clicks like a machine gun is usually a flat or weak battery.
Could be a worn bit of motor - try hitting the large cylinder, a jolt can knock it to a good bit (you will still need a new starter as it will do it again).
Also check all connections as SteveFMk1 suggested with spade connector. Three in total, the spade one and two rings on a bolt near to it. The spade is from ignition (which seems to be working) and the two ring connectorrs are earth and battery power.
Recon starters are about £50-£70.