Summary: In this paper we consider the attainability of the optimal constant \(S(N,p,s,\mu)\) associated to the fractional Hardy-Sobolev type embedding which is defined as \[ S(N,p,s,\mu): = \inf\limits_{u\in\dot{W}^{s,p}(\mathbb{R}^N)\backslash\{0\}} \frac{\int_{\mathbb{R}^N} \int_{\mathbb{R}^N}\frac{|u(x)-u(y)|^p}{|x-y|^{N+ps}}dxdy-\mu\int_{\mathbb{R}^N}\frac{|u|^p} {|x|^{ps}}dx}{\big(\int_{\mathbb{R}^N}|u|^{p^{\ast}_s}dx\big)^{\frac{p}{p^{\ast}_s}}}, \] where \(s\in(0,1)\), \(p >1\) and \(N>ps\), \(0\leq\mu<\mu_H\), the latter being the best constant in the fractional Hardy inequality on \(\mathbb{R}^N\), \(p_s^{\ast} = \frac{Np}{N-ps}\) is the fractional critical Sobolev exponent. The technique that we use to prove the existence of extremals for \(S(N,p,s,\mu)\) is based on blow-up analysis argument combined with a variational method. Further, as an application of the inequality, we prove an existence result for the critical fractional \(p\)-Laplacian equation with Hardy potential and involving continuous nonlinearities having quasicritical growth.