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The most common problem with these later version furnaces is the OEM control board which is made in China. Crap! Just replaced on for a customer yesterday. I would get an after market board made by Dinosaur Electronics as they are far superior to the OEM boards and have an outstanding warranty. If you know the make and model of your furnace I can supply a copy of the Service Manual and guide you in the repair. Now............if the "click" you heard was in the ceiling box at the A/C then it may be that control box is bad. I need the make and part number off that in order to direct you further. We can do some testing to rule out this as a cause. Please tell me if you have an outside inspection panel on the furnace. That makes getting to the control board much easier. All follow ups here are included in our fee and fully expected.