<p>This paper designs a C-FISTA-type Krasnosel’skiĭ–Mann iterative algorithm to find a common fixed point of a family of quasi-nonexpansive operators in Hilbert spaces. The proposed algorithm involves the Krasnosel’skiĭ–Mann iteration with two momentum terms and a correction term. We establish that the sequence of iterates generated by the proposed algorithm converges weakly to a common fixed point of the family of quasi-nonexpansive operators. We also demonstrate that a linear convergence rate is achieved under standard assumptions. We apply our results to solve optimization problems that can be converted to fixed-point problems. Numerical results for TV-based denoising problems and image in-painting demonstrate the superiority of our proposed algorithm over the Krasnosel’skiĭ–Mann iteration and inertial Krasnosel’skiĭ–Mann iterations reported in the literature.</p>