We explore the area of fairness in clustering from the different perspective of modifying clusterings from existing algorithms to make them fairer whilst retaining their quality. We formulate the minimal cluster modification for fairness (MCMF) problem where the input is a given partitional clustering and the goal is to minimally change it so that the clustering is still of good quality and fairer. We show using an intricate case analysis that for a single protected variable, the problem is efficiently solvable (i.e., in the class P) by proving that the constraint matrix for an integer linear programming (ILP) formulation is totally unimodular (TU). Interestingly, we show that even for a single protected variable, the addition of simple pairwise guidance (to say ensure individual level fairness) makes the MCMF problem computationally intractable (i.e., NP-hard). Experimental results on Twitter, Census and NYT data sets show that our methods can modify existing clusterings for data sets in excess of 100,000 instances within minutes on laptops and find as fair but higher quality clusterings than fair by design clustering algorithms.