The click is the relay kicking in but not completing the circuit. You didn't say what kind of car you have. Fords mount the relay on the inner fender. GM relay is part of the solenoid on top of the starter. When I have that problem I tighten the starter connections. If that doesn't work I take the starter cable off, clean it and the nut with sandpaper and put it back together. It's worked every time. On Fords the relay connection needs to be cleaned too. On a GM car you may need to clean all the connections on the solenoid.
AL