When a child has been coughing for a long time and does not have a stuffy nose or fever, it could be due to a slight mucus drainage from the back of their throat, coming from their nose. One helpful technique to try is nasal hygiene, which involves using a syringe (without a needle) and saline or distilled water to reduce the amount of mucus draining from the nose to the throat. This technique can often help alleviate the coughing and congestion.