Energy Harvesting and Energy Efficient (EEH) Cognitive Radio Networks (CRNs) is one of the key technologies to meet the next generation wireless network demands for high energy and spectrum efficiency. EEH-CRNs can enable self-sustaining green communications by reducing the energy cost and harvesting the ambient energy sources while capitalizing the idle spectrum simultaneously. In this paper, we first propose a hybrid EH-SU model to harvest energy from both renewable sources, e.g. solar, and ambient radio frequency signals. A general hybrid cooperative spectrum sensing (CSS) scheme is then considered with and without energy half-duplex (EHD) constraint which prevents SUs from charging and discharging the battery at the same time. As an alternative to common homogeneity assumption, we propose a heterogeneous EEH-CSS scheme to exploit heterogeneous sensing and reporting channel characteristics of SUs. After formulating the energy state evolution under stochastic energy arrivals, a convex myopic EEH-CSS policy optimization framework is then developed to jointly obtain the optimal harvesting ratio, sensing duration and detection threshold of each SU to maximize the total achievable throughput subject to collision and energy-causality constraints. Obtained results show that the proposed heterogeneous approach delivers %45 and %230 more throughput than the homogeneous one with and without EHD constraint, respectively. Furthermore, if the EHD constraint is mitigated, proposed heterogeneous approach provides %400 and %240 more throughput than the EHD constrained homogeneous and heterogeneous EHE-CSS schemes, respectively.