Summary: This paper deals with existence and multiplicity of positive solutions to the following class of nonlocal equations with critical nonlinearity: \[ \begin{cases} (-\mathit{\Delta})^s u -\gamma\dfrac{u}{|x|^{2s}}=K(x)\dfrac{|u|^{2^*_s(t)-2}u}{|x|^t}+f(x) \quad\text{in}\quad\mathbb{R}^N,\\ \qquad\qquad\qquad\quad u\in \dot{H}^s(\mathbb{R}^N), \end{cases} \] where \(N > 2s, s \in (0, 1), 0 \leq t < 2s < N\) and \(2^*_s(t):=\frac{2(N-t)}{N-2s}\). Here \(0 < \gamma < \gamma_{N,s}\) and \(\gamma_{N,s}\) is the best Hardy constant in the fractional Hardy inequality. The coefficient \(K\) is a positive continuous function on \(\mathbb{R}^N\), with \(K(0) = 1 = lim_{|x| \rightarrow \infty }K(x)\). The perturbation \(f\) is a nonnegative nontrivial functional in the dual space \(\dot{H}^s(\mathbb{R}^N)'\) of \(\dot{H}^s( \mathbb{R}^N)\). We establish the profile decomposition of the Palais-Smale sequence associated with the functional. Further, if \(K \geq 1\) and \(\| f \|_{(\dot{H}^s)'}\) is small enough (but \(f \not\equiv 0)\), we establish existence of at least two positive solutions to the above equation.