  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
$\sqrt{s} = 13$ TeV during LHC Run2 (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.
