Sorry about your frustration regarding your child’s persistent cough. Your child most likely has a form of cold allergy, which many other kids get; coughing when it is very cold, especially at night.
Realize this is different from allergic rhinitis where they have a cough, sneeze, have a blocked or runny and itchy teary eye among other signs when exposed to irritants like dust, perfumes, and even cold.
You do not need to give any antibiotic for your child unless the child gets a fever and clear signs of infection, in which case visiting a doctor to evaluate the child and write a prescription would be the ideal next step. Cod liver oil has no proven role or benefit in this case. The children usually outgrow this allergy by age five-seven years.
For now, clothe the child warmly at night; in coveralls, or long-sleeved warm top with trousers and socks because kids don’t like a blanket to cover them in the night.
Syrup cetirizine is the other accompanying option. Give 5ml before bed, every night on very cold nights, or if a cough becomes worse in the night. So you can use the cetirizine syrup as on and off, as you wait for the child’s immunity to grow to adult immunity, which occurs at around five-seven years.
Some children even outgrow this allergy as early as age three-four years. Your child otherwise is OK. Don’t worry and don’t give those supplements. Instead, feed the child on a balanced and normal diet. Enjoy and celebrate motherhood.