Hello, my name is Dr. Kara and I have over 20 years of experience as a veterinarian.
I understand your concern for this little one.
A newborn pup that has milk coming out of his nose and is getting weaker is very concerning. The most likely cause of this is a cleft palate or an opening between the mouth and nose that did not properly form and close while in utero due to a congenital defect or a toxin exposure in the womb.
When these little ones try and nurse they have difficulty getting good suction because they cannot form a tight seal and the milk can come out the nose as well as going down the esophagus. They have to work much harder to eat then the other puppies so they often have difficulty gaining weight and get weak due to inadequate nutrition. They can also get secondary infections in their nose from having milk in their nose.
Here are some pictures of puppies with a cleft palate:
Here is a comparison between a normal and affected puppy:
You may be able to save this puppy if you can learn to tube feed him so he can get good nutrition. But it is possible that he may have other congenital defects along with the cleft palate which may make survival difficult. And he will need surgery later to fix his defect. He should not be bred as he could pass this on to his puppies.
I recommend that you take him to be examined by a veterinarian. They can teach you how to tube feed him and look for other congenital defects. If he is too sick to survive they can humanely euthanize him for you.
Please let me know if you have any further questions.