you are most likely correct IF the click is the relay. It could also be the busted switch
Is your steering wheel set up with the two horn buttons like they had on most early airbags, or do you just hit the center of the horn like normal cars?
Most cars have 2 horns, your civic is no exception. One's on the left of the bumper, the other's on the right. If you're not getting sound from BOTH of them, that kinda makes me think horn switch.
You may end up wanting to track down an electrical service manual for your car and go after it yourself unless you determine it's in the wheel - shops like to charge HUNDREDS of dollars for electrical diagnostic work.