If you try to brazen it out and continue regardless it will eventually stop you one way or another. I had one winter when I tried to persist with racing and developed pneumonia-like symptoms, raspy chest, unable to talk in sentences, night sweats, and generally feeling awful. You can actually make yourself very ill - there is no such thing as 'mild asthma'.
It is a very good idea to take two puffs of your reliever about 15 minutes before you go out, and always carry it with you.
Do cover your mouth. The best thing to use is a buff (www.buffwear.co.uk), just the original microfibre ones. They are absolutely brilliant. You can breathe quite well through them at low and moderate intensities. As I said you will be unlikely to be able to do intense sessions outdoors in any case.
Take much longer than you normally would to warm up. Don't just go out of the door from warm, dry air to cold damp air, running hard. This will certainly bring on symptoms!
When thoroughly warmed up you can perhaps take the buff off and just leave it round your neck - even this is a great help. It seems to warm the air going down the throat before it reaches the lungs.
First of all you should know that you can continue to enjoy your running even if you have asthma. You will find at most just a small percentage may drop from your performance.
BUT it sounds like, as I do, you need to avoid racing and intense sessions outside in the cold. I have come to accept that I cannot race from the beginning of Nov to the end of Feb. breathing in cold air brings on the asthma syptoms.
With asthma the secret is to get it controlled. This will not be done by just using the reliever ('blue') inhaler. You will need a preventer as well which you should use religiously twice a day. It is usual practice to start with a high dose and gradually reduce to a maintenance dose that is the minimum needed to avoid symptoms.
I'm comfortable with two SNODs in one day, which is a start, and what I intend to do again in 2017. I could build up from there, I suppose, and do three in two days the following year, then four in two days, and so on.
The trouble with my scenario is that they would all be done in darkness, apart from the official race. This would add an extra dimension of difficulty, or enjoyment, depending on outlook!