We consider cooperative communications with energy harvesting (EH) relays, and develop a distributed power control mechanism for the relaying terminals. Unlike prior art which mainly deal with single-relay systems with saturated traffic flow, we address the case of bursty data arrival at the source cooperatively forwarded by multiple half-duplex EH relays. We aim at optimizing the long-run average delay of the source packets under the energy neutrality constraint on power consumption of each relay. While EH relay systems have been predominantly optimized using either offline or online methodologies, we take on a more realistic learning-theoretic approach. Hence, our scheme can be deployed for real-time operation without assuming acausal information on channel realizations, data/energy arrivals as required by offline optimization, nor does it rely on precise statistics of the system processes as is the case with online optimization. We formulate the problem as a partially observable identical payoff stochastic game (PO-IPSG) with factored controllers, in which the power control policy of each relay is adaptive to its local source-to-relay/relay-to-destination channel states, its local energy state as well as to the source buffer state information. We derive a multi-agent reinforcement learning algorithm which is convergent to a locally optimal solution of the formulated PO-IPSG. The proposed algorithm operates without explicit message exchange between the relays, while inducing only little source-relay signaling overhead. By simulation, we contrast the delay performance of the proposed method against existing heuristics for throughput maximization. It is shown that compared with these heuristics, the systematic approach adopted in this paper has a smaller sub-optimality gap once evaluated against a centralized optimal policy armed with perfect statistics.