Hello! My son has had a cough for over a week now. There's no runny no…

Hello! My son has had a cough for over a week now. There's no runny nose anymore, but I clean his nose at night. Sometimes when he eats, I hear something like a wheeze or a cough, it's hard to tell. Yesterday, the pediatrician checked again and found his lungs clear and his throat not red anymore. She prescribed Augmentin antibiotic. I think that since his throat is not red, maybe the problem is in his nose. Maybe there's irritation there and he coughs because of it, even though he still coughs when his nose is clear. Who gave him Augmentin? Should I give it to him?