Since the cough and cold is persisting you have to rule out infection / allergy.
Allergy due to cold is one of the common causes of cough and cold in children. if cough and cold increases at night and morning then most probably it would be cold allergy.
Sometimes there is superadded infection along with cold allergy. doctors visit is important so as to rule out infection and allergy to cold.
Some homemade remedies which you can try at home are,
Giving Honey to the child while coughing, warm water for drinking, warm milk. This home based remedies decreases throat dryness due to cold and decreases coughing.
Keep the child warm, make the child wear formal clothes, keep the room warm with a room heater.
Next Steps
meet your nearest pediatrician