A little olive oil added, uncooked, to your dog’s daily food helps with winter itchies and flaky skin. For small dogs, start with a quarter to a half teaspoon twice daily. (Not too much. You don’t want to cause tummy troubles.)
Give the oil supplement for a couple of weeks, and you should see marked improvement in skin texture and greatly reduced scratching, if the scratching was initially caused by dry skin. If it doesn’t clear up, talk to your vet.