Pop the button off with the plate and screws still installed, disconnect the wire from the horn plate, if it still goes off replace the relay.
If it doesn't go off make sure the button fits correctly and isn't hitting the wheel or nut anywhere.

However, if it's still going off and you know the relay is good, check this:
-The spring that goes over the wheel nut should only be touching the plastic/formica side of the button plate. If the spring touches metal the horn goes off.

The three screws that hold the plate in place have a stepped shank to set the height of the heads and provide proper clearance for the switch. Unless the replacmeent part is wrong or they are the wrong screws you should be able to tighten them w/o the horn plate touching the wheel. If not, do what BurBert said.
The screws should not be contacting the metal portion of the plate either.

1956 GMC 370 dump " 'Tater "
1970 VW Volksrod "the Black Bomber"
2007 Chevy Avalanche
2020 Jeep Cherokee Trailhawk,
2005 Harley Davidson FLHTCUI

I dig all cars, old and new, whether they were hammered out of American iron, German steel, or Japanese tin cans. Being unable to appreciate them all is missing out on a world of great things.
But thats just MY opinion! :P