We consider a communication system with energy harvesting at a receiver for which the processing energy is the bottleneck. We propose using hybrid automatic retransmission request (HARQ) with soft combining to reduce the processing energy and improve the throughput under limited receiver energy. In this protocol, the receiver keeps requesting additional redundancy in order to increase the gap between transmission rate and the overall accumulated mutual information (so-called capacity gap), which in turn reduces the processing energy. We compare the performance of two HARQ schemes: (1) incremental redundancy (IR) HARQ, which incrementally increases the code-length by sending additional coded symbols (parity symbols) in each retransmission and (2) Repetition-HARQ, which simply repeats transmitting the same coded message (codeword) in response to the retransmission requests, and applies maximum ratio combining (MRC) at the receiver. In these systems, the decoding energy is a decreasing function of the capacity gap but an increasing function of the code-length. The IR-HARQ protocol yields a better capacity gap, but increases the code-length, while Repetition-HARQ offers less improvement in the capacity gap, but does not increase the effective code-length. Thus, contrary to systems without an energy constraint on receiver processing in which IR-HARQ always performs better, here, depending on the system parameters, repetition-HARQ can outperform IR-HARQ.