Many, many, of us keep our dogs home during the day. However, your pup is 9 weeks old and really belongs with its mother until 12 weeks. If you can have someone check on your pup throughout the day or dogsit until her health improves that would be great for your pup.
If not, I didn't see where your vet diagnosed the pup? If possible, take a video of your pup coughing so the vet can see. Also, coughing can indicate many things including heart issues. Please make sure your vet is checking for everything and if need be get an ekg and/or ultrasound.
Good Luck I hope it gets better soon.
