Are you checking for voltage at the connecter with it unplugged from the
horn? It needs to be unplugged for a proper reading. One wire is a ground
(black) and the other wire is power ( green). If you connect your meter
across both wires you should have 12 volts when you press the horn button.
If you are getting half a volt with the connecter unplugged from the horn
then you have a problem. It can be on the power or the ground side of the
cicuit. A quick check would be to ground your meter at the battery negative
post and check for voltage at the power side of the horn connecter. If you
now have 12 volts you have a bad ground. If you still have only half a volt
the problem is on the power side. I would remove the relay and inspect the
terminals for corrosion. You can also check the relay socket to see if you
are getting 12 volts to the relay. There will be 4 terminals in the relay
socket, you should have 12 volts on 2 of them, they are both fed by the same
wire.