Respiratory infections in puppies during transitions to a new home are fairly common. I compare it to kids starting back to school, it would be rare to visit a classroom that had no coughing, sneezing or runny nosed kids. Stress and immature immune systems are the common denominators in both situations. Many respiratory infections of puppies are mixed infections meaning they have a viral and bacterial component. We need to remember that antibiotics are not effective against viruses but fortunately viral respiratory infections are usually self limiting so they do not last for long periods.
As long as your puppy eats well and feels good he will get well. While a single 10 day course of antibiotics defeats many infections it is no surprise that some puppies require treatment for longer periods. If your puppy has improved on the antibiotic it should be continued, if he has not improved your vet may change to a different antibiotic. It is sometimes necessary to continue therapy for several weeks. In my experience clavamox is often disappointing.
Adding the use of a nebulizer or a moist bathroom environment for 20 minutes 2-3 times daily if your puppy will tolerate it often is helpful. I also like using expectorants to help your puppy to bring up any fluid or debris created by the infection. Guaifenesin is a safe expectorant available as an over the counter syrup. Some veterinarians suggest using cough suppressants but while there may be some artificial improvement in the frequency or intensity of the cough it seems that overall the cough lasts longer. We need to recognize that a cough is part of the body's attempt to eliminate the residue left by infection in the airway, suppressing this natural response does not make sense to me.
Please keep me informed.