Welcome to HCM,
Your son has cold and breathing problem and cough, he may be having allergic rhinitis
as he responded to inhaler so kindly consult a Chest Physician and investigate with CBC,Serum IGE level, a X Ray chest,Pulmonary function test and if required Allergy
Along with inhaler he may need antibiotics,anti histaminics,and anti inflammatory cortcosteroids inhalation or oral depending on his requirement.
If he has nasal congestion
there are Intra nasal sprays to be taken after consultation with a ENT Surgeon.