All an injury does to a wolf is cause it to deal half the damage it normally deals to prey and enimies in explore. (For example, your wolf will deal 15 damage if it normally deals 30.) It won't kill them.
However, an injury will stay until you heal it. You can heal injuries with one of the three types of healing potions: red heals infected wounds, yellow heals broken tails, and green heals injured paws. You get healing potions by crafting them out of the herbs you find in explore or buying them out of people's market dens. You can also use a robin feather to heal injuries, those heal any type and brings your wolf's health to 100. You get those by chasing off the birdwatcher in the forest or buying them from people's market dens.
Hope this helps. ^^
Spring Month: 3 
Night 
Heavy Rain Moon: 
