We generalize the { ${\rm M}$-estimator} put forward by Catoni in his seminal paper [C12] to the case in which samples can have finite $α$-th moment with $α\in (1,2)$ rather than finite variance, our approach is by slightly modifying the influence function $\varphi$ therein. The choice of the new influence function is inspired by the Taylor-like expansion developed in [C-N-X]. We obtain a deviation bound of the estimator, as $α\rightarrow 2$, this bound is the same as that in [C12]. Experiment shows that our generalized ${\rm M}$-estimator performs better than the empirical mean estimator, the smaller the $α$ is, the better the performance will be. As an application, we study an $\ell_{1}$ regression considered by Zhang et al. [Z-Z] who assumed that samples have finite variance, and relax their assumption to be finite {$α$-th} moment with $α\in (1,2)$.