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 root 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.