Regional co-location pattern mining is a branch of spatial co-location pattern mining, which is used to discover co-location patterns that prevalently co-occur in local regions. The regional co-location patterns can reveal the association relationships among spatial features in the local regions. However, in practice, the association relationship between certain features is asymmetrical, and some features play a key (core) role in the relationship, as central position. To explore the regional co-location patterns with core feature (called regional core patterns (RCPs)) as well as their prevalent regions, this paper presents a regional partition method based on the influence of core feature. Additionally, a regional core pattern mining (RCPM) algorithm is proposed to mine the RCPs and unveil their spatial distribution resulting from the arrangement of core feature. In the regional partition stage, we propose a partition criterion that takes into account the influence of core feature. This criterion aims to comprehensively address the spatial distribution relationship between core instances and their neighboring non-core instances, ensuring the partition process is rational and complete. In the RCPM stage, we propose a core-based nearest affiliation measure method to assess the neighbor relationship between core and non-core instances, which can consider the fact that there is competition between spatial instances of the same spatial feature. And, to effectively calculate pattern prevalence and quickly identify RCPs, we present a data structure called Core_Hash to store the influence relationship between core and non-core instances. Extensive experimental evaluations and analyses are conducted on synthetic and real-world datasets. Compared to the existing algorithms, our proposed algorithms yield more reasonable and comprehensive RCPs and demonstrate good efficiency and scalability.