This paper develops a control framework for a network of energy harvesting nodes connected to a Base Station (BS) over a multiple access channel. Due to fluctuations in energy availability and, possibly, energy outages, the number of nodes attempting channel access is random and varies over time. Thus, each node must carefully adapt its access probability to the network state to optimize network performance. In order to reduce the complexity of network control, a lightweight and flexible design framework is proposed where energy storage dynamics are replaced by dynamic average power constraints at each node, induced by the time correlated energy supply. The BS adapts the access probability of the 'active' nodes (those currently under a favorable energy harvesting state) so as to maximize throughput. The resulting policy takes the form of access probability as a function of the local energy harvesting state and number of active nodes. The structure of the throughput-optimal policy is analytically derived for the genie-aided case of non-causal knowledge of the number of active nodes. Inspired by it, a Bayesian estimation approach is presented for the more practical scenario where the BS estimates the number of active nodes. The proposed scheme is shown to outperform by 20% a scheme in which the nodes operate based only on local state information, and to be robust against the impact of energy storage dynamics at a fraction of the complexity.