The behaviour of multi-agent learning in competitive settings is often considered under the restrictive assumption of a zero-sum game. Only under this strict requirement is the behaviour of learning well understood; beyond this, learning dynamics can often display non-convergent behaviours which prevent fixed-point analysis. Nonetheless, many relevant competitive games do not satisfy the zero-sum assumption. Motivated by this, we study a smooth variant of Q-Learning, a popular reinforcement learning dynamics which balances the agents' tendency to maximise their payoffs with their propensity to explore the state space. We examine this dynamic in games which are 'close' to network zero-sum games and find that Q-Learning converges to a neighbourhood around a unique equilibrium. The size of the neighbourhood is determined by the 'distance' to the zero-sum game, as well as the exploration rates of the agents. We complement these results by providing a method whereby, given an arbitrary network game, the 'nearest' network zero-sum game can be found efficiently. As our experiments show, these guarantees are independent of whether the dynamics ultimately reach an equilibrium, or remain non-convergent.