Cliché with determination and eagerness. The angry soldier attacked the enemy with a vengeance. Bill ate all his dinner and gobbled up his dessert with a vengeance.
With great violence or energy; also, to an extreme degree. For example, The cottage was filthy and Ruth began cleaning with a vengeance, or December has turned cold with a vengeance. This expression was first recorded in 1533. Also see with a will.