Probably a broken piece in the center of the steering wheel.  Twist the horn piece counterclockwise to remove it.  Compare the parts inside to some on CJPony catalog page to see if the plastic piece is broken or if something is missing.

I was considering buying a new horn ring since the chrome is not in good shape on mine.  Would that fix it or does it not come with any parts, just the horn ring itself?

Not to steal the thread, along the same lines
If the steering shaft is too close the horn will honk itself as you turn the steering wheel, usually at the most awkward moment!! I adjusted the shaft down a bit and it only honks when I want.

Funny, my horn hasn't worked since I fitted my 15" Wood steering whewheel my dash and headlights flicker when I press the horn but NO Horn working 

im sure its something to do with the horn  button contact on the wheel.

jimmy O

Mine is the fake GT350 wood one that Kentucky mustang sell same as what's on the Cobra Kit cars,It's not an aftermarket copy of anything that was ever on the mustang like the wood Gt wheel

i know its in my horn button connection app cause when I remove and reinstall the spring clips on horn button Spark and the horn tries to work,both horns are ok and work when power is put to them.

jimmy O