Envisioning the potentials of energy harvesting technology and the improved spectrum reuse by joint utilization of overlay and underlay modes, this paper studies the throughput performance of a novel cognitive radio network (CRN) scenario with a mobile energy-harvesting secondary transmitter (ST). The hybrid overlay-underlay scheme allows the secondary users to access the spectrum even when the primary signal is detected. We are the first to partition the unit area into three parts for secondary users: overlay mode area, underlay mode area, and harvesting zone. Then, we propose a metric to classify the CRN into the spectrum-limited state and the energy-limited state, and accordingly maximize the throughput through the monotonicity analysis of throughput and collision probability. The secondary throughput is maximized under the energy constraint and collision constraint. Moreover, we quantitatively discuss the impacts of underlay mode transmission on the classification of network states and the corresponding optimal spectrum sensing, respectively. We find that with a relatively small detection threshold, ST transmits the considerable amount of packets in underlay mode, while it transmits few packets in overlay mode. Theoretical results are validated by simulations, and our findings shed light on the design and operation of mobile energy-harvesting CRNs.