This paper develops a control framework for a network of energy harvesting nodes connected to a Base Station (BS) over a multiple access channel. The objective is to adapt their transmission strategy to the state of the network, including the energy available to the individual nodes. In order to reduce the complexity of control, an optimization framework is proposed where energy storage dynamics are replaced by dynamic average power constraints induced by the time correlated energy supply, thus enabling lightweight and flexible network control. Specifically, the BS adapts the packet transmission probability of the "active" nodes (those currently under a favorable energy harvesting state) so as to maximize the average long-term throughput, under these dynamic average power constraints. The resulting policy takes the form of the packet transmission probability as a function of the energy harvesting state and number of active nodes. The structure of the throughput-optimal genie-aided policy, in which the number of active nodes is known non-causally at the BS, is proved. Inspired by the genie-aided policy, a Bayesian estimation approach is presented to address the case where the BS estimates the number of active nodes based on the observed network transmission pattern. It is shown that the proposed scheme outperforms by 20% a scheme in which the nodes operate based on local state information only, and performs well even when energy storage dynamics are taken into account.