How do i get rid of cough and phlegm?

First of all, the antibiotic you took did not give you the sinus infection, but the fact that you did not FINISH taking all of the medicine could have. When you have an infection and take anitbiotic but don't finish it, only the weakest bacteria are killed off. By stopping the medicine before you have finished all of it, you allow the strongest and most resistant bacteria to keep living. This is why bacteria are becoming harder to treat. Even if you have gone to the Dr. before, your mom needs to take you again. If he prescribes medicine YOU MUST FINISH all of it, EVEN IF YOU FEEL BETTER. If you take the wrong antibiotic for your condition it can potentially make what you have worse. If she absolutely will not take you to the dr, drink lots of water, hot herbal tea (no caffeine) with lemon and/or honey, try taking Mucinex for the mucous or phlegm and a cough suppressent to help your throat irritation from coughing so much. Use a humidifier, get lots of rest, stay warm. The other thing is that dairy products and sugar are mucous producing, so try to stay away from them for a few days. Hope you feel better!