Hi Marcus
Sounds like your GTR has been fitted with a 5 wire ignition switch from the 175/200 models (these are more readily available and cheaper).
As far as I can tell, the 350 has an extra circuit, which swiches in the 3rd phase of the alternator when the lights are in use for extra power. This uses the brown wire from the alternator which the switch connects to the white wire to the rectifier.
Your horn should be connected to the battery using blue wire as shown in the GTR wiring diagram. It needs battery power to work properly. There may be a spare blue socket on the switched live coming from your switch, if not splice it in.
Unfortunately the wiring diagram for the 175 in the manual on this site is not colour coded so it is difficult to compare the 175/350 circuits.
Try this - connect horn as above. Connect the brown and white wires together (near the rectifier where they meet). This effectively bypasses the ignition switch and gives a useful battery charge boost.
Alternatively, buy a 6wire GTR switch. Richard shows 1 in stock but Kerry Pagenkopf (facebook site) may have also have a new or good used one. I bought one from him recently.
Brian.
Edit. From a colour coded 175 wiring digram that Mike Dunham sent me (Thanks mike), the 175 switch shown does have a brown wire and shows 7 connections from the switch. So I,m not sure now what switch you have. Mikes wiring diagram also varies from the one shown in the on-site manual and may be a later 175 wiring update which does include brown/white wires for switching in the 3rd alternator phase. However, I believe the suggestions I made above should still work if you have previously run the bike with the fitted 5wire switch.