
At 4 months old your pup should be fine having a play in the snow.
It depends a lot on the breed too, a 4 month old malamute would obviously cope better in the cold than a 4 month old yorkshire terrrier.
I had a litter of 6 week old pups out in the snow yesterday - only for a few minutes, they had a little peep & a play & then went back inside.