  Scalar-tensor theories are well studied extensions of general relativity that
offer deviations which are yet within observational boundaries. We present the
time evolution equations governing the perturbations of a nonrotating
scalarized neutron star, including a dynamic spacetime as well as scalar field
within the framework of such scalar-tensor theories. We employ a theory that
allows for a massive scalar field or a self-interaction term and we study the
impact of those parameters on the non-axisymmetric $f$-mode. The time evolution
approach allows for a comparatively simple implementation of the boundary
conditions. We find that the $f$-mode frequency is no longer a simple function
of the star's average density when a scalar field is present. We also evaluate
the accuracy of different variants of the Cowling approximation commonly used
in previous studies of neutron star oscillation modes in alternative theories
of gravity and demonstrate that it can give us not only qualitatively correct
results, but in some cases also good quantitative estimates of the oscillations
frequencies.
