We address learning Nash equilibria in convex games under the payoff information setting. We consider the case in which the game pseudo-gradient is monotone but not necessarily strictly monotone. This relaxation of strict monotonicity enables application of learning algorithms to a larger class of games, such as, for example, a zero-sum game with a merely convex-concave cost function. We derive an algorithm whose iterates provably converge to the least-norm Nash equilibrium in this setting. {From the perspective of a single player using the proposed algorithm, we view the game as an instance of online optimization}. Through this lens, we quantify the regret rate of the algorithm and provide an approach to choose the algorithm's parameters to minimize the regret rate.