NO. Fuel pump relay controls the fuel delivery. You should NOT be hearing ANY clicking sounds when starting. If you have a manual tranny, you have a starting system that includes: a starter relay, a clutch safety switch, ignition switch, starter, battery.
On my \'91 Integra, I by-passed the normal starting system by running 2 separate 12-guage wires to the dash (thru the firewall, mounted along the firewall over across to the starter and to the under-hood fuse panel). I connected 1 wire to the under hood fuse panel (for power), and the other wire directly to the starter relay. Then, at the dash I connected both wires to a high-quality "momentary-on/off" toggle switch - this is my starter switch. I turn the ignition key to when the dash lights come on, then flick the switch, and BINGO the car always starts right up.
BUT, this is somewhat unsafe because it doesn\'t use the clutch safety switch - which is nothing more than a spring-loaded switch on the arm of the clutch pedal connected to the ground. IF you are using the standard starting system, you MUST engage the clutch for the starting system to make contact with the ground in order to engage the starter. So, check this switch first for a good connection to ground. It usually fails first before anything else. If it checks out, THEN check the starter relay, located waaaaaaay up underneath the dash, right in the middle. To get at the starter relay, you\'ll have to remove the glove box, and plastic & metal kick panels on the passenger side. The relay is located way up underneath the dash, next to the Blower housing. You\'ll need a long socket extension to reach it. Someone at Acura/Honda decided this was the best place for it. Boy would I like to whack him with a 2x4.
See this diagram for reference - the starter relay is #15.