  If coupled \emph{feebly} to the Standard Model bath, a dark matter can evade
the severe constraints from the direct search experiments. At the same time,
such interactions help produce dark matter via the freeze-in mechanism. The
freeze-in scenario becomes more interesting if one also includes the thermal
masses of the different particles involved in the dark matter phenomenology.
Incorporating such thermal corrections opens up the possibility of dark matter
production via forbidden channels that remain kinematically disallowed in the
standard freeze-in setup. Motivated by this, we investigate such freeze-in
production of the dark matter in a minimally extended $U(1)_{L_\mu-L_\tau}$
framework that remains consistent with the recent muon $(g-2)$ data. Here, the
role of the dark matter is played by the scalar with a non-trivial charge under
the additional symmetry $U(1)_{L_\mu-L_\tau}$. This scalar dark matter obtains
a thermally corrected mass at high temperatures for a not-so-small
self-coupling. We show that the thermal correction to the dark matter mass
plays a significant role in the dark matter phenomenology.
