This paper presents a search for dark matter in the context of a two-Higgs-doublet model together with an additional pseudoscalar mediator, a, which decays into the dark-matter particles. Processes where the pseudoscalar mediator is produced in association with a single top quark in the 2HDM+a model are explored for the first time at the LHC. Several final states which include either one or two charged leptons (electrons or muons) and a significant amount of missing transverse momentum are considered. The analysis is based on proton–proton collision data collected with the ATLAS experiment at s=13 TeV during LHC Run 2 (2015–2018), corresponding to an integrated luminosity of 139 fb ^{- 1}. No significant excess above the Standard Model predictions is found. The results are expressed as 95% confidence-level limits on the parameters of the signal models considered.