This sounds like what happened with my bigscreen tv. But I could see arcing, and smell ozone.
Apparently what happens is that the energy required to charge the gas in the crt is shunted before it gets to where its supposed to go. There's energy in the order of kilovolts and it can kill you should it discharge through your body, so be careful when poking around inside there.
I assume your monitor is just like my tv: there is a wire going from the crt to some box near where the power enters the monitor. What I did was to coat this box with a thick coating of that orange gasket maker stuff for cars. This stopped the clicking and the power went to the picture tube like it was supposed to. This repair has lasted for 18 months now. I guess the coating wears out after awhile.
edit>> disclaimer: should you kill or injure yourself by doing this, I hereby absolve myself of any liability

I think there's a way to discharge what I think is a capacitor, but I'm not sure how.