Hi welcome to the health care magic
Here
bronchitis should be ruled out as cough and mucus present ....
You need to investigated by...
-Auscultation
-Chest x ray
-
Spirometry etc...
Spirometry will able to assess severity and will help in further management of
asthma as per Spirometry finding
Your CBC done and if AEC more than 400 than fexofenedine course for 15 days can be Prescribed...
You should be searched for suspected allergen and for that allergic testing done...
For maintainance in asthma usually formetrol and
budesonide containing inhaler enough...
The cough can occur in asthma and with improvement in asthma (by
montelukast and fexofenedine course) cough will subside
If bronchitis found in x ray than only antibiotic needed
With keeping my suggestion in mind consult pulmonologoist for your examination and work up accordingly
Hope your concern solved