Traffic offloading through heterogenous small-cell networks (HSCNs) has been envisioned as a cost-efficient approach to accommodate the tremendous traffic growth in cellular networks. In this paper, we investigate an energy-efficient dual-connectivity (DC) enabled traffic offloading through HSCNs, in which small cells are powered in a hybrid manner including both the conventional on-grid power-supply and renewable energy harvested from environment. To achieve a flexible traffic offloading, the emerging DC-enabled traffic offloading in 3GPP specification allows each mobile user (MU) to simultaneously communicate with a macro cell and offload data through a small cell. In spite of saving the on-grid power consumption, powering traffic offloading by energy harvesting (EH) might lead to quality of service (QoS) degradation, e.g., when the EH power-supply fails to support the required offloading rate. Thus, to reap the benefits of the DC-capability and the EH power-supply, we propose a joint optimization of traffic scheduling and power allocation that aims at minimizing the total on-grid power consumption of macro and small cells, while guaranteeing each served MU