Strong AGN heating provides an alternative means for the disruption of cluster cool cores (CCs) to cluster mergers. In this work we present a systematic Chandra study of a sample of 108 nearby ($z<0.1$) galaxy clusters, to investigate the effect of AGN heating on CCs. About 40% of clusters with small offsets between the BCG and the X-ray centre ($\le50$ kpc) have small CCs. For comparison, 14 of 17 clusters with large offsets have small CCs, which suggests that mergers or sloshing can be efficient in reducing the CC size. Relaxed, small CC clusters generally have weak radio AGNs ($P_{1.4\rm GHz}<10^{23}$ W Hz$^{-1}$), and they show a lack of systems hosting a radio AGN with intermediate radio power ($2\times10^{23}
Comment: 17 pages, 10 figures, Accepted for publication on MNRAS