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This is not a good sign. Often a puppy will "gasp" like this right before it goes into shock. The puppy getting chilled is also an issue. Since the pup is in poor shape I want to start by giving you information on resuscitation and fading puppy syndrome. There are many reasons why your pup may not be doing well that include serious worm infestation, flea infestation causing anemia, herpes virus, and even pneumonia from milk aspiration or congenital health issues. It is sometimes referred to as fading puppy syndrome. This site has information on fading puppy syndrome.
This site goes over ways of helping save fading puppies.
The pup needs to be kept warm. Mom's will often push a pup away from the warmth of the litter and the pup becomes chilled and develops pneumonia and dies as a result. For pups at risk, it is better to have a heating bad under a towel and put the pup where it can move on and off of the heated par of the towel if they get too hot. It will help keep them from getting chilled. It can also help a bit if they have aspirated a small amount of liquid into the lungs.
Unfortunately, part of breeding is that some pups don't make it. That might be the case with this pup especially if it has become chilled multiple times and is now gasping for breath. Read how to get a pup breathing again here:
It is for newborns but the same techniques are applied for young pups as well.
I hope this information is helpful to you. If you would like any additional information or have more questions please don’t hesitate to press the reply to expert or continue conversation button so I can address any issues you still have . If you do find this helpful, please take this opportunity to rate my answer positively so I am compensated for my time.