Days after respiratory infection I still have postinfectious cough and still producing a little phlegm is this normal when will it go away?

6 weeks. The cough associated with bronchitis can last for up to 6 weeks. You don't need antibiotics unless you are feverish or short of breath (and other symptoms, then you should see your doctor). You would just need medication to treat the cough.
Possibility... You may be having some bronchospasm associated with airway inflammation, which can occur post-infection. The treatment includes a bronchodilator inhaler and a steroid inhaler so see your doctor to see if this is what is going on and, if so, your doctor can prescribe this therapy to help clear these symptoms. Good luck!
Maybe. There are lots of things that can cause a persistent cough. Some will go way on their own like colds, viral infections etc. Others will not go away such as chronic sinusitus, allergies etc. If you are still coughing, call your family doctor.