Just use AutoHotKey with this script and press F12 for show/hide Putty (Use AutoHotKey Window Spy for know your SupperPutty class, because it could be change):
F12:: IfWinExist ahk_class WindowsForms10.Window.8.app.0.37a0697 { IfWinActive { WinMinimize } else { WinActivate } } else { Run C:\<path>\SupperPutty.exe }