Issue with horn

Don't overlook Pishtas post; all the info is right there.
Relays can make pinging noises, but it is usually covered up by the thing they are relaying,lol.The pinging means the signal side of the relay is working.
Another thing to consider is that it only takes a tiny bit of electricity to turn on a test light, whereas it takes much,much more electricity to operate the device that is being activated;the horn in this case.
Before you do anything, just make sure the body is grounded to the battery, and that the column is grounded to the chassis.There should be a little short wire from somewhere in the area behind the driver's side headlight or the driver's side of the rad core-support, going to the battery negative post.And there should be a jumper on the column going around one of the mounting posts,connecting the column to the post, which is connected to the chassis through several fasteners. Without that jumper the ground circuit would have to find the battery through a rather torturous path.So try these two first.
Then if still a no-honk,then I think you need to try and pull some amps through that relay.
To do this you will need to prove the relay is good;all 4 terminals.
Bring the horn directly to the relay,ground it and jumper it to the relay load hot-side (which goes to the battery).If it does not honk on the supposed-to-be-hot load side, then the issue is between the hot connection and the battery;go fix it.
But,If it does honk there, jumper the horn to the load-side relay-side and hit the horn button.A honk now, indicates a problem between the relay and the horn's underhood location, OR it COULD mean the signal side is defective,and for a no-honk, we need to dig deeper.leave the horn jumpered there for a bit.
To find out which, jumper the ground side of the signal circuit,and turn the key on. A no honk now, means still, that the issue could be at the underhood location, and still could be at the relay, to its power side.Turn the key off.
To find out which, leave the signal side ground jumper installed, and jumper the other signal side terminal to a known good battery supply, like the previously proven or repaired load hot-side terminal which we already proved goes to the battery. A honk now, indicates that the signal side of the relay itself is good, but that it's power supply is faulty.Go fix it.After fixing the power side, you need to recheck the horn button side, by removing the jumper there and restoring the connection to the horn button. Press the button.A no-honk now means the circuit from there to the horn button is faulty.
Once the signal side of the relay is proved to be good, and the circuit is good,including the horn button, and if it then still does not honk from the underhood location, then the wire from there to the relay, by process of elimination, is proven to be bad. Go find the problem area and fix it.
The bulkhead connector is a popular trouble spot.The relays last for decades upon decades.It's case does not require grounding. None of its circuits are fused,AFAIK. It does however pull power thru the fusible link. But if that is faulty, the car would neither crank nor run,nor would the headlights work.
Good luck.
I know it sound complicated, but it's not. It is just a process of jumpering connections and/or eliminating those 4 circuits. Pishtas drawing is really nice; it clearly shows were everything is coming and going.