We consider energy-harvesting cognitive radio (CR) systems with finite batteries. The energy-constrained secondary users (SUs) can be coordinated to enhance both the primary user (PU) detection and the opportunistic utilization of the PU spectrum. We employ centralized cooperative spectrum sensing, and the optimal sensors are chosen from among the SUs with varied received PU signal power levels and energy-arrival rates. The finite-horizon partially observable Markov decision process (POMDP) is applied to obtain the optimal cooperation among the SUs for sensing and access to maximize throughput with the available energy, while satisfying the PU detection constraint. The sensing-access policy decides whether to sense, the set of sensing SUs, the sensing detection threshold, and the SU that accesses the spectrum opportunities. The optimal policy achieves an optimum tradeoff between SU transmissions, preventing energy shortage and obtaining sensing information for the future gains.