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 Run2 (2015-2018), corresponding to an integrated luminosity of 139 fb$\mathrm{<mrow\; data-mjx-texclass="ORD">-1</mrow>}$. 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.