I have had a sore throat for a week now along with a cough it gets worse when I smoke and I coughed up green mucus. Sinus infection or throatt cancer?

Stop smoking. One week of throat pain is not an indication that you have throat cancer. But if you continue to smoke, it is possible that some day that may occur. You are only 18 and the symptoms are more consistent with an upper respiratory tract infection, which can be made worse by smoking. Smoking at your age is bound to cause increased infections and decreased immunity. Try to stop.
Two parts. Part one: Go to your local, regular MD and discuss the symptoms and see if they can help you. It is statistically most likely an infection of some sort. Part two: Clearly you are worried about cancer and admit in your question you are smoking. It might be reasonable to guess you know you should quit. It is hard to quit..try to get help from the doc too. Having that support really helps success. .