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Recovery from uncomplicated infectious tracheobronchitis (kennel cough) usually takes about 2 weeks. Clavamox is widely used to treat secondary infections. I agree with your concern about stopping antibiotics while there is a lot of coughing. Generally, if secondary infections and pneumonia occur, dogs get a fever and have a poor appetite.
You do not want to stop a productive cough with drugs, but there are some things that might help. A humidifier can be useful. You can also take the pup into the bathroom while running hot water in the shower to increase the humidity.
Pediatric cough syrup containing guaifenesin can be used. Guaifenesin is an expectorant and loosens thick mucous allowing it to move up the trachea and get coughed out. Guaifenesin often comes combined with cough suppressants (dextromethorphan). The combo is OK because it doesn't generally stop coughing. Do NOT get products with Tylenol/acetaminophen or other ingredients. A reasonable dose for a 10 pound puppy is about 1/4 teaspoon (1.25 ml) 3-4 times a day of the pediatric (not adult) syrups.
The roundworms are a concern. They have a very interesting life cycle. Encysted larva are activated during pregnancy and are passed to the unborn pups across the placenta. They also transmit via milk. Some of the reactivated larva pass to the mother's intestine and mature. The mature worms lay eggs that can infect the pups and other dogs that sniff feces or surfaces contaminated with fecal material. Ingested eggs hatch in the intestines and larva penetrate the intestines entering the body. They migrate through the body and can cause significant damage. Eventually they get into the lungs and are coughed up and swallowed. If the number of migrating larva are large, the cough can be severe. It is a good idea to continue treating every couple weeks for at least 3-4 doses. The treatment kills adults in the intestines, but not the larva. All of the once a month heartworm preventatives are labeled to treat roundworm.
It is a good idea to check the pups temperature and have the lungs listened to again.
Let me know if you have follow up questions.