In order to improve the performance of sending or not sending quantum key distribution (SNS-QKD), an SNS-QKD protocol based on a heralded pair-coherent source (HPCS) is proposed, named HPCS-SNS-QKD protocol. In the protocol, a weak coherent source (WCS) is replaced by HPCS. HPCS has lower ratio of vacuum state pulse than WCS. Hence the single photon error bit rate can be effectively reduced in HPCS-SNS-QKD. The relationship between the key generation rate and secure transmission distance is derived, and the expression of the key generation rate under limited pulses is given. The numerical simulation results show that the proposed HPCS-SNS-QKD protocol can exceed the upper limit of single-photon key transmission distance. Compared with WCS-SNS-QKD protocol, HPCS-SNS-QKD protocol has longer transmission distance and higher key generation rate and can endure larger misalignment error. For limited pulses in practical experiment, HPCS-SNS-QKD always has a higher key generation rate.