Since the symptoms are due to cool weather prefer giving home remedies. But if you find that the child is uncomfortable or restless due to the cough then better visit a doctor

NO,do NOT self medicate,even for cold and cough. If mild,just give steam inhalations for cold and cough and decongestant nasal drops for nasal block or increased cough on lying down. Use paracetamol for fever if required. Recurrent cold and cough could be due to allergy,rather than infections. If not improving OR accompanied by other symptoms like wheezing or breathlessness or dullness and refusal to feed,consult your paediatrician for physical examination and treatment.