It helps to have an understanding of how the gas control works. The gas control valve operates a solenoid-operated valve that delivers gas to the burners when your thermostat calls for heat.
The pilot flame should always be burning, and heating the thermocouple bulb that is situated adjacent to the pilot flame. The thermocouple feeds a signal via a thin copper tube to the gas control valve, indicating that the pilot is burning and that it's OK for the control valve to open and deliver gas to the burners.
In my ~35 years with a furnace similar to yours (it was replaced about 8 years ago with an electronic ignition furnace), I occasionally found that either:
The pilot light's orifice had become dirty and was not producing an adequate flame to heat the thermocouple,
Or, more often, the thermocouple bulb had become slightly corroded and was failing to sense the pilot's heat
The pilot light's orifice can be removed and cleaned.
The thermocouple's bulb can also be removed and cleaned. I typically used a bit of bronze wool (I hate steel wool!).
As to the clicking sound that you report, I don't know what that might be. Perhaps it's the solenoid in the gas control valve attempting to open gas flow to the burners, but it's prohibited from doing so by the thermocouple's failure to sense a burning pilot.
Note that the tube from the thermocouple does not contain any gas or electrical signal. It functions on the basis of heat at the thermocouple's bulb causing the inert gas inside the tube to expand.