In this work, communication over a time-correlated point-to-point wireless channel is studied for an energy harvesting (EH) transmitter. In this model, we take into account the time and energy cost of acquiring channel state information. At the beginning of the time slot, the EH transmitter, has to choose among three possible actions: i) deferring the transmission to save its energy for future use, ii) transmitting without sensing, and iii) sensing the channel before transmission. At each time slot, the transmitter chooses one of the three possible actions to maximize the total expected discounted number of bits transmitted over an infinite time horizon. This problem can be formulated as a partially observable Markov decision process (POMDP) which is then converted to an ordinary MDP by introducing a belief on the channel state, and the optimal policy is shown to exhibit a threshold behavior on the belief state, with battery-dependent threshold values. Optimal threshold values and corresponding optimal performance are characterized through numerical simulations, and it is shown that having the sensing action and intelligently using it to track the channel state improves the achievable long-term throughput significantly.