  This paper presents a search for dark matter, $\chi$, using events with a
single top quark and an energetic $W$ boson. The analysis is based on
proton-proton collision data collected with the ATLAS experiment at $\sqrt{s}=$
13 TeV during LHC Run 2 (2015-2018), corresponding to an integrated luminosity
of 139 fb$^{-1}$. The search considers final states with zero or one charged
lepton (electron or muon), at least one $b$-jet and large missing transverse
momentum. In addition, a result from a previous search considering
two-charged-lepton final states is included in the interpretation of the
results. The data are found to be in good agreement with the Standard Model
predictions and the results are interpreted in terms of 95% confidence-level
exclusion limits in the context of a class of dark matter models involving an
extended two-Higgs-doublet sector together with a pseudoscalar mediator
particle. The search is particularly sensitive to on-shell production of the
charged Higgs boson state, $H^{\pm}$, arising from the two-Higgs-doublet
mixing, and its semi-invisible decays via the mediator particle, $a$: $H^{\pm}
\rightarrow W^\pm a (\rightarrow \chi\chi)$. Signal models with $H^{\pm}$
masses up to 1.5 TeV and $a$ masses up to 350 GeV are excluded assuming a
tan$\beta$ value of 1. For masses of $a$ of 150 (250) GeV, tan$\beta$ values up
to 2 are excluded for $H^{\pm}$ masses between 200 (400) GeV and 1.5 TeV.
Signals with tan$\beta$ values between 20 and 30 are excluded for $H^{\pm}$
masses between 500 and 800 GeV.
