It is pretty common for scratchy throat symptoms or even cough to persist for a while after having a common cold or flu. This is because the damage done to small nerves in the back of the throat from the viral infection can be sufficient enough that it requires a while to heal up, even up to several weeks. As long as the symptoms continue to improve, there is not much you need to do, and you can take cough drops or anti inflammatory medications for the symptoms.
On the other hand, if your throat starts to become more sore without a return of cold or flu symptoms, then it is possible that you have developed a second problem, such as a throat infection. Throat infections from bacteria like strep throat are relatively common, and they may be even more common in the period right after a cold or flu, since the cold or flu may set you up for the second infection.
If your are worried, or the symptoms worsen, you should go see your primary care doctor
. He or she will be able to perform a simple throat swab to make sure you are not dealing with a throat infection.