The horns are earthed to the frame.
Ok
The existing horn wiring connects to the relay (possibly the positive and earth are the wrong way round on the relay - if it makes a difference).
I dont think that would matter, this is the two coil connections of the relay.
The relay gets a direct feed from the battery and direct out to the horns (possibly these input/output are the wrong way round on the relay and need to be swapped - if it makes a difference).
This would not matter, this is the switch connections of the relay, I would recomend going through a fuse though.
Although - I did have to 'persuade' the dash wiring loom to fit inside the dash when reassembling (it made a tight dog leg). But I can't see that would be the cause because then the problem wouldn't just be isolated to the horn use - it would manifest at other times as well.
I think your problem must be in there, guessing you put the relay in there, the horn button on the Lh bar I think operates by grounding the horns,(now your relay) this wire on your relay must grounding out the other circuits.
Check over the connections and if thats no good I would disconnect the new horns from the relay and also the live from the battery, see if the relay clicks when you press horn button, and of corse if you short out the two wires you disconnected new horns should sound. may help you find what side the short could be on.
Happy hunting.