  We study the scalarized charged black holes in the Einstein-Maxwell-Scalar
(EMS) theory with scalar mass term. In this work, the scalar mass term is
chosen to be $m^2_\phi=\alpha/\beta$, where $\alpha$ is a coupling parameter
and $\beta$ is a mass-like parameter. It turns out that any scalarized charged
black holes are not allowed for the case of $\beta \le 4.4$ with
$q=0.7(M=0.5,Q=0.35)$ because this case implies the stable
Reissner-Nodstr\"{o}m (RN) black holes. In the massless limit of $\beta\to
\infty$, one recovers the case of the EMS theory. We note that the unstable RN
black hole implies the appearance of scalarized charged black holes. The other
unstable case of $\beta>4.4$ with $q=0.7$ allows us to obtain the
$n=0,1,2,\cdots$ scalarized charged black holes for $\alpha(\beta)\ge
\alpha_{\rm th}(\beta)$ where $\alpha_{\rm th}(\beta)$ represents the threshold
of instability for the RN black hole. Furthermore, it is shown that the $n=0$
black hole is stable against radial perturbations, while the $n=1$ black hole
is unstable. This stability result is independent of the mass parameter
$\beta$.
