  Results from a wide range of searches targeting different experimental
signatures with and without missing transverse momentum
($E_{\mathrm{T}}^{\mathrm{miss}}$) are used to constrain a Two-Higgs-Doublet
Model (2HDM) with an additional pseudo-scalar mediating the interaction between
ordinary and dark matter (2HDM+$a$). The analyses use up to 139 fb$^{-1}$ of
proton-proton collision data at a centre-of-mass energy $\sqrt{s}=13$ TeV
recorded with the ATLAS detector at the Large Hadron Collider between
2015-2018. The results from three of the most sensitive searches are combined
statistically. These searches target signatures with large
$E_{\mathrm{T}}^{\mathrm{miss}}$ and a leptonically decaying $Z$ boson; large
$E_{\mathrm{T}}^{\mathrm{miss}}$ and a Higgs boson decaying to bottom quarks;
and production of charged Higgs bosons in final states with top and bottom
quarks, respectively. Constraints are derived for several common as well as new
benchmark scenarios within the 2HDM+$a$.
