the click you can hear sounds to me like the starter solenoid pulling in. ( I dont think your 1200 would have any relays or anything else that goes click, if the car is pretty much standard)
It could be the contacts under the starters solenoid cap like others have said! this is very common in starters.
If its a "bosch" startermotor you can get the cap off and you can buy a kit which is a new cap and contacts. Most probly its not a Bosch starter though!

My first bet for your problem is the brushes inside the starter need replacing as they are now too short.