Why does the relay and control board keeps clicking the igniter doesn't come on, and furnance doesn't work?

It should only do that a few times then lock out necessitating reset by cycling the power at the furnace switch or at the t-stat. More than likely your hot surface ignitor is defective. Sequence should be something like this: Call for heat, combustion air fan initiates and blows for 30 seconds during a 'pre-purge' condition, then a relay clicks to energize the hot surface ignitor. If it does not ignite the flame within a few seconds, usually around 5 or 6 seconds, the ignitor relay clicks again and drops it out.

The flame sensing rod has to see flame to rectify the ground signal to the circuit board very quickly or else raw gas would continue to dump into the heat chamber. After the HSI relay drops out on flame failure the combustion air fan continues to run for a post purge cycle to purge the heat exchanger of any unburnt fuel before the sequence is initiated again.

Unplug the ignitor molex plug. It'll be within a few inches of the HSI itself. Insert a voltage test meter set to read 120 volts into the molex plug going TO the ignitor. Re-establish the call for heat, wait for a cycle of the relay clicks and steadily watch the meter. If the ignitor is bad, you should see 120 volts at the connection you are plugged in to AFTER the 30 second pre-purge. Watch closely, you may only have a 6 second window to see it because it will drop it out on flame failure.

Of course there are other issues that could be possible, but this is the most likely. If you do determine the ignitor is bad and you obtain a replacement DO NOT TOUCH the portion of the ignitor that glows (to ignite the gas) during assembly !!!! The oil from your skin will shorten the life expectancy TREMENDOUSLY.