Results from a wide range of searches targeting different experimental signatures with and without missing transverse momentum (E _T ^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 s=13 TeV recorded with the ATLAS detector at the Large Hadron Collider during 2015–2018. The results from three of the most sensitive searches are combined statistically. These searches target signatures with large E _T ^miss and a leptonically decaying Z boson; large E _T ^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 and new benchmark scenarios in the 2HDM+a.