I have asthma that is triggered by cold air (and by certain foods). The air doesn't have to be frigid, if it's chilly, that's enough to start me coughing as soon as I breathe in through my mouth. What has really helped me is this: I wear a lightweight buff or balaclava over my lower face, covering my nose and mouth. The fit is loose enough to create a pocket of warm humid air around my face, and to let fresh air in around the sides of my nose. I'm not really drawing breath through the fabric itself. My helmet straps keep the buff from falling down; it's not tight enough to stay in place on its own. If it's not too cold I sometimes leave my nose uncovered and just put the buff over my mouth, as breathing through my nose doesn't cause coughing fits as readily. It might be worth trying; you don't lose much if it doesn't work for you!