I will start this out saying I am not a vet but I just got an 11 week old yorkie baby and she was having some coughing problems last week.
"Kennel Cough" is really a generalization for what could be a number of upper respiratory infections or viruses and just like humans get the flu shot for a specific strand of the flu, dogs get shots for specific strands of viruses.
A vet can not ask your dog exactly where the problem is and how it feels in their throat so they generalize and will prescribe antibiotics. If the puppy has a virus, just like in humans antibiotics will not slow down the virus it needs to run its course. A puppy will have a weakened immune system from fighting the virus and the antibiotics can help stop any other major respiratory problems from happening.
With all of this said, just like a young child there are a few things that can help ease these coughs. A humidifier in the room with the puppy will help loosen the cough. I have heard from other sources as well as posted above that honey helps ease the cough. I have given my puppy a little bit of robitussin CF when hers got bad, it was approximately 1/8-1/10 of a teaspoon, I would not do this without consulting your veterinarian but it did help her sleep.
It is not uncommon for your puppy to cough at night when trying to go to bed or when they wake up, think about when you have a cough and what happens to your throat at night. Last, I would not worry yet about the cough subsiding, this can take time depending on exactly what is causing the cough. As long as there aren't other infections in the body it should be ok.
With all of that said! I still get upset when Bella is coughing only now I tell her to cough it up and get it out!