Had a cold a month ago now I'm left with a tickle in my throat that causes me to cough aggressively daily. It Stopped for a week and came back. Why?

Ongoing drainage. If your sinuses are draining and you are swallowing or aspirating into the lungs you will cough. See your doctor to determine if your proper diagnosis and if you need prescription treatment. If you have already been treated for strep you may have a secondary viral infection or gotten reinfected. See your doctor and feel better soon.