The glow plugs are little heating elements. The resistance in the circuit varies as the heating element gets hot. The controller is calibrated for a resistance range that is commiserate with all the plugs working. If they are not all working, the resistance is out of range and you get the short cycle click thing happening because the controller isn't seeing what it expects. Typically, when a glow plug fails it goes from around 1 ohm of resistance to open (the element is broken and it is no longer a complete circuit). Typically you test them either with a test light or a multimeter. With a test light, unplug the wire off the top (otherwise they will read through the other plugs) and connect the power for the test light to the battery +. Touch the top of the glow plug. If it's good, the test light should light up. If not, it won't. For the multimeter, hook one lead to the battery negative and do the same thing with the other lead. You should see 1-2 ohms. If it's a lot more than that, check the connections. If the connections are good, the GPs may be getting tired.
Hope that helps.