Hi. My 2.5 Year Old Daughter Has Wet Cough For

Hi. My 2.5 year old daughter has wet cough for over a month now with slighly runny nose (think yellow discharge). The Pediatrician checked her twice and said that her lungs are clear and it must be coming from her throat. He is not a big fan of syrups and said that her cough should go by itself soon. She is really suffering with coughing attacks especially when she falls asleep. Sometimes I feel like it is getting better and then it becomes worse. Can this type of cough be treated with Mucolyte syrup? Any advise from your side will much appreciated. Thank you.
Thank you.