Does my son have asthma?

Hi everyone

I'm just posting for a bit of advice. My son is 5 years old and has been coughing at night for about a month or so. I took him to the GP about 3 weeks ago who listened to his chest and said one lung was'nt as clear as it should be , so put him on Co-Amoxiclav (antibiotics) They didn't clear the cough and today when we got up the weather was really really cold, and he's been coughing literally all day. It's a strange cough, sounds like he is straining sort of. Anyway, I took him to the Emergency doctor who gave him a Salbutamol inhaler which seems to lessen the coughing, although it hasnt gone completely

Anyone have any ideas?