In this paper, we study wireless networks where nodes have two energy sources, namely a battery and radio frequency (RF) energy harvesting circuitry. We formulate two optimization problems with different objective functions, namely maximizing the sum throughput and maximizing the minimum throughput, for enhanced fairness. Furthermore, we show the generality of the proposed system model through characterizing the conditions under which the two formulated optimization problems can be reduced to the corresponding problems of different known wireless networks, namely, conventional wireless networks (battery-powered) and wireless powered communications networks (WPCNs) with only RF energy harvesting nodes. In addition, we introduce WPCNs with two types of nodes, with and without RF energy harvesting capability, in which the nodes without RF energy harvesting are utilized to enhance the sum throughput, even beyond WPCNs with all energy harvesting nodes. We establish the convexity of all formulated problems which opens room for efficient solution using standard techniques. Our numerical results show that the two types of wireless networks, namely WPCNs with only RF energy harvesting nodes and conventional wireless networks, are considered, respectively, as lower and upper bounds on the performance of the generalized problem setting in terms of the maximum sum throughput and the maxmin throughput. Moreover, the results reveal new insights and throughput-fairness trade-offs unique to our new problem setting.