Thank you for your patience. These symptoms in such a young dog are definitely a worry - particularly the excessive sleeping. There are a few things that could be going on here, but a gagging type cough in a young puppy is commonly associated with canine infectious cough (aka kennel cough). Hopefully this little one is up to date with vaccinations, including the kennel cough vaccine, as if she isn't, this infectious cough is even more likely. Thankfully, if she does indeed have kennel cough, then it is relatively easily treated with antibiotics, so long as it is diagnosed and treated asap. If left, then pneumonia can potentially develop as a consequence.
Other potential causes for this cough and lethargy include anything from parasites (lung or heart worm for example), so please make sure that your girl is up to date with all of her deworming treatments, and again if she isn't - please try and do this asap with a reliable product from your local vet. We also have to consider a congenital issue that your girl may have been born with here, and these can include anything from a hypoplastic trachea (small windpipe), to a congenital heart issue.
Because she has been quite lethargic with a low appetite, I would definitely encourage you to follow up with your local vet on this today if you can. Given her age and her symptoms, it is most likely that this is kennel cough, but she will need to have a full exam and temperature check. Because this is a potentially infectious issue, please keep her away from other dogs in public and let the vet know her symptoms before you turn up to the clinic. Commonly dogs will be examined in the car or car park to prevent potentially spreading the disease. Treatment is simple (antibiotics) and very effective, but do get her seen sooner rather than later to get a diagnosis and ensure this isn't due to something else.
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