This paper studies online power control for battery-limited point-to-point energy harvesting communications over slow block-fading channels. A linear-policy-based approximation is developed for the relative-value function in the Bellman equation of the power control problem. This approximation leads to two fundamental parameterized clipped affine policies: an optimistic policy derived from a certainty-equivalence-type approximation and a robust policy derived from worst-case analysis. For independent and identically distributed energy arrivals and channel states, two families of power control schemes are developed based on the optimistic clipped affine (OCA) and robust clipped affine (RCA) policies, respectively. The proposed adaptive RCA policy based on reinforcement learning (RCA-RL) is further extended to address four scenarios with contextual information: one-step energy lookahead, one-step channel lookahead, one-step joint energy-channel lookahead, and Markov energy arrivals. Extensive simulation results show that the proposed schemes provide a favorable tradeoff between computational complexity and performance. The adaptive RCA policy based on the maximin optimal linear-policy-slope approximation (RCA-OLA-A) and the RCA-RL scheme achieve the best overall performance, while the RCA policy based on the maximin optimal linear policy (RCA-OL) is the best-performing closed-form policy. In particular, RCA-OLA-A, RCA-RL, and the aforementioned RCA-RL extensions achieve less than 2% performance loss relative to the optimal policy across a range of scenarios, consistently outperforming the considered benchmark approaches, including generic reinforcement learning baselines. The RCA-OL policy also performs well with less than 4% performance loss.

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