Unknown ticking sound on bad ite power supply
First, a warning: Power supplies like this one contain high-voltage capacitors that will hold a potentially deadly charge even after you unplugged it from the mains. So if you unplug the power supply and then start to work on it, you might still get a pretty dangerous electric shock.

The problem with this power supply is most likely caused due to the controller chip's feedback/bootstrap circuitry not working.

Most power supplies have a controller chip (and switching transistors) on the primary side which creates the waveforms needed to drive the main ferrite transformer. This chip needs to be powered as well, so most manufacturers of such power supplies put an additional low-voltage winding on the transformer which will power the controller chip. This creates a chicken-and-egg problem, however: The controller chip can't start before it has power, but it doesn't get power before it starts and drives the transformer. In order to solve this problem, there's a large-value resistor that slowly charges a low-voltage capacitor from the ~300V line voltage. When this capacitor has enough voltage, the controller IC can start the power supply using the energy in the capacitor and then sustain its own operation using the auxiliary transformer winding.

If the auxiliary winding does not work for some reason, the chip will use up the capacitor's energy and then shut down again which results in a clicking noise from the transformer. This repeats every time the capacitor has reached the start-up voltage of the controller chip, causing rhytmic clicking.

If I had to take a guess, I'd say there's a dry solder joint or a dead diode somewhere in the thing.