Green’s Dictionary of Slang

cutty gun n.

also cuttie gun
[Scot./northern UK dial. cutty-gun, a short, i.e. ‘cut-down’ gun; image of the penis as a weapon]

the penis.

[Scot]Burns ‘Andrew an’ his Cuttie Gun’ in Merry Muses of Caledonia (1965) 147: As Andrew fodgel’d wi his airse, / And fir’d at me the cuttie gun.
[Scot] in Laughing Songster 14: [song title] Andro and his Cutty Gun.
[UK]Farmer & Henley Sl. and Its Analogues.