Most existing neural video codecs (NVCs) only extract short-term temporal context by optical flow-based motion compensation. However, such short-term temporal context suffers from error propagation and lacks awareness of long-term relevant information. This limits their performance, particularly in a long prediction chain. In this paper, we address the issue by facilitating the synergy of both long-term and short-term temporal contexts during feature propagation. Specifically, we introduce a Long-term Temporal Context Gathering (LTCG) module to search the diverse and relevant context from the long-term reference feature. The searched long-term context is leveraged to refine the feature propagation by integrating into the short-term reference feature, which can enhance the reconstruction quality and mitigate the propagation errors. During the search process, how to distinguish the helpful context and filter the irrelevant information is challenging and vital. To this end, we cluster the reference feature and perform the searching process in an intra-cluster fashion to improve the context mining. This synergistic integration of long-term and short-term temporal contexts can significantly enhance the temporal correlation modeling. Additionally, to improve the probability estimation in variable-bitrate coding, we introduce the quantization parameter as an extra prior to the entropy model. Comprehensive evaluations demonstrate the effectiveness of our method, which offers an average 11.3\% bitrate saving over the ECM on 1080p video datasets, using the single intra-frame setting.