Hi there and thanks for your question.
I'm sorry to hear that this pup is having a hard time!
Often, when mom pushes away a pup and won't feed it it is because she just knows that there is something wrong with the pup.
From what you are describing I am concerned that the pup may have a cleft palate. Sometimes this is visible on exam if you open up his mouth and look at the roof of his mouth. Compare it to the other pups. Do you see a space in the middle?
If so, this is a condition that can be corrected with surgery but it is not a simple surgery to do. While some dogs can live with a mild cleft palate, unfortunately most end up being euthanized.
Also, it is not uncommon for a dog with cleft palate to have other defects which often are not readily visible on examination.
The above is given for information only. Although I am a licensed veterinarian, I cannot legally prescribe medicines or diagnose your pet's condition without performing a physical exam. If you have concerns about your pet I would highly advise contacting your regular veterinarian.
I would still put cleft palate first on the list. Sometimes there is only a very small hole that can be seen under anesthetic. However, if this is the case then there is a good chance that if we can get this pup through the next few weeks and onto solid food that he could pull through.
You may want to consider tube feeding him. You can find information on how to do this here:
Tube feeding does come with risks, but I think it is probably the only way that you are going to be able to be able to get calories into this pup. You will likely need to have a vet visit to get the equipement you need to do the tube feeding.
In the meantime, continue to try nursing, or you can try syringing some of the milk in if you have a syringe.
Keep in mind, however, that if this is a cleft palate there is a chance that there are other problems with him. And if this is not a cleft palate then there is likely some type of anatomical problem such as a problem with his swallowing reflex.
If you manage to get him to survive to 3 weeks of age you can start offering some watered down puppy food then. (We usually start at 4, but if he will take it at 3 you can make an exception because of his troubles with taking in liquid.)
I hope he pulls through!