With nothing else going on, you should have full battery voltage at the relay, even if there is corrosion or dirty connections. Connect an Ohmmeter between the power/battery connection and the power (not grounded) terminal of the battery. Anything over 0.1 Ohm, you have a problem. The other way to test it is to leave your voltmeter hooked to the Bat terminal of the relay and observe the voltage drop when you push the horn button. If it drops more than half a volt, you have too much resistance somewhere in the circuit.
Clean and tighten all connections to the relay. Any sign of green copper corrosion at any connector probably means that the wire is corroding inside the jacket and will need to be replaced. Take the relay loose from its mounting and clean the contact to be sure it has a good ground. If the relay is mounted on the horn bracket, check the ground between the horn bracket and the body to which it is mounted.
These horns are sensitive to low voltage.