  Observations of white dwarfs in dark matter-rich environments can provide
strong limits on the strength of dark matter interactions. Here we apply the
recently improved formalism of the dark matter capture rate in white dwarfs to
a general model in which dark matter interacts with the white dwarf ion
components via a light scalar mediator. We compute the dark matter capture rate
in the optically thin limit in a cold white dwarf from the globular cluster
Messier. We then estimate the threshold cross-section, which significantly
varies as a function of the light scalar mediator mass $m_\phi$ in the range of
$0.05\, m_\chi<m_\phi<m_\chi$ and becomes constant when $m_\phi>m_\chi$. We
also show that the bounds obtained from the dark matter capture in a white
dwarf from the globular cluster Messier 4 are complementary to direct detection
experiments and particularly strong in the sub-GeV regime.
