No, a suppository is not recommended for a 16-month-old with a wet cough caused by excessive secretions. It is better to reduce the amount of secretions in the throat by minimizing nasal drips. This can be done by following the technique described in the provided document. Ensure you stay at home with your child for 14 days if they have a cough, and contact a healthcare professional if they develop a fever or if the fever persists for more than three days.