The clicking comes from the piggyback starter motor solenoid so you should not be surprised that burnt out. What usually happens though is either a near flat or time expired battery is not able to provide enough current. The otherwise perfect solenoid drops out just after starter is engaged as the current draw causes the knackered battery volts to drop very low. Battery volts then recover and the process repeats in very short order, several times a second hence clicking. Although less likely it can be caused by poor battery or earth strap connections.
Bosch used to provide the solenoids as a separate part although replacement involved soldering. Nowadays a failed one means a complete new starter motor as solenoid is essentially an integral part.
It occurs to me you may be calling the starter relay a solenoid. The former is a standalone device one step further up the starting chain and could also cause the issue although unlikely. Always worth carrying a spare as it's a simple plug in replacement. Ditto fuel pump relay, they are often the same.
