There has been much interest recently in developing fair clustering algorithms that seek to do justice to the representation of groups defined along sensitive attributes such as race and gender . We observe that clustering algorithms could generate clusters such that different groups are disadvantaged within different clusters. We develop a clustering algorithm, building upon the centroid clustering paradigm pioneered by classical algorithms such as k -means, where we focus on mitigating the unfairness experienced by the most-disadvantaged group within each cluster. Our method uses an iterative optimisation paradigm whereby an initial cluster assignment is modified by reassigning objects to clusters such that the worst-off sensitive group within each cluster is benefitted. We demonstrate the effectiveness of our method through extensive empirical evaluations over a novel evaluation metric on real-world datasets. Specifically, we show that our method is effective in enhancing cluster-level group representativity fairness significantly at low impact on cluster coherence.