In this paper, we develop an analytical framework to evaluate the network throughput and traffic offloading efficiency in heterogeneous cellular networks (HCNs) with multiple energy-harvesting personal cells. The network consists of a tier of power-grid connected base-stations (BSs) and multiple tiers of energy-harvesting (personal) small cell access points (EH-SAPs). Each EH-SAP is personally owned by a priority user and becomes active only when its priority user requests communication and when its residual battery energy is sufficient to support the transmission. The energy arrival at each EH-SAP is modeled as a Bernoulli random process and the battery energy dynamics are modeled as a discrete time Markov chain. The network throughput is derived and its dependence on the energy arrival probability, cell association bias, and transmission power are examined based on the analysis. We show that, when the transmission power of EH-SAPs is sufficiently small, network throughput can be increased by increasing the density of EH-SAPs and by offloading more traffic to them. Our results offer insights on the design of spectral-efficient HCNs with EH-SAPs.