We consider the (k)-clustering problem with (\ell_p)-norm cost, which includes (k)-median, (k)-means and (k)-center, under an individual notion of fairness proposed by Jung et al. [2020]: given a set of points (P) of size (n), a set of (k) centers induces a fair clustering if every point in (P) has a center among its (n/k) closest neighbors. Mahabadi and Vakilian [2020] presented a ((p^{O(p)},7))-bicriteria approximation for fair clustering with (\ell_p)-norm cost: every point finds a center within distance at most (7) times its distance to its ((n/k))-th closest neighbor and the (\ell_p)-norm cost of the solution is at most (p^{O(p)}) times the cost of an optimal fair solution. In this work, for any (\epsilon>0), we present an improved ((16^p +\epsilon,3))-bicriteria for this problem. Moreover, for (p=1) ((k)-median) and (p=\infty) ((k)-center), we present improved cost-approximation factors (7.081+\epsilon) and (3+\epsilon) respectively. To achieve our guarantees, we extend the framework of [Charikar et al., 2002, Swamy, 2016] and devise a (16^p)-approximation algorithm for the facility location with (\ell_p)-norm cost under matroid constraint which might be of an independent interest. Besides, our approach suggests a reduction from our individually fair clustering to a clustering with a group fairness requirement proposed by Kleindessner et al. [2019], which is essentially the median matroid problem [Krishnaswamy et al., 2011].