Bit Of Advice Needed

Pure guess but would it be some kind of breather or recirculation hose? What year bike is it? I could look in my Haynes manual but not for a few days as I'm on holiday in Scotland.

 

Is it not the feed pipe hole from the airbox for the air injection system?
Looking at this video particularly from the 15.56 time point:-

Technical info: "The secondary air injection system (AIS) routes fresh air from the airbox to the exhaust system in order to aid the combustion of unburned fuel in the exhaust."

I did the same work on my Speedmaster (2005), removing the air injection system. I've still got the complete air injection system sitting in a box in my garage.

 

Hi Joe hope you are keeping well, it is for a secondary air induction that controls fuel that remains in the exhaust, without this you can get flames out of the rear exhaust due to ignition from any unburned fuel particles, exactly like you see on say the BSB race bikes, it may look spectacular but in reality it causes more issues on road going machines. Ride safe all from an ageing Rocker.