Can playing around in snow for a few minutes hurt a child with asthma?

Maybe. If the child has cold air induced asthma then playing outside can cause symptoms. The asthma symptoms can be prevented by covering the nose and mouth with a scarf and taking an albuterol inhaler 2 inhalations, 10 minutes before going outside.
Snow OK 4 asthma. Any issues with snow are more about the cold air. Breathing cold air can trigger the asthma process in some people, so it is best if you can have the child breathe through a scarf or neck warmer at least until they are well warmed up. A pre-activity inhalation of albuterol can also help if the play will be vigorous. Many elite athletes, including snow sports, have asthma, so play on!