yes, you likely caused a short in the horn circuit messing with the contact, the button should supply a ground to the relay only when you push the button - as daddy points out, you can reconnect what you disconnected and unplug the horn wire where it exits the column to see if that's the problem
"why is the Amp gauge affected by the horn relay?"
because you have something wired wrong in the circuit involving the relay, the regulator, the ammeter and the light switch, which are all interconnected, or you also disconnected a wire from the regulator .... look at the wiring diagram in your shop manual and check your wiring [
this one will work if you don't have a manual]
Bill
