  A search for events with one top quark and missing transverse momentum in the
final state is presented. The fully hadronic decay of the top quark is explored
by selecting events with a reconstructed boosted top-quark topology produced in
association with large missing transverse momentum. The analysis uses 139
fb$^{-1}$ of proton-proton collision data at a centre-of-mass energy of
$\sqrt{s}$=13 TeV recorded during 2015-2018 by the ATLAS detector at the Large
Hadron Collider. The results are interpreted in the context of simplified
models for Dark Matter particle production and the single production of a
vector-like $T$ quark. In the absence of a significant excess with respect to
the Standard Model expectations, 95% confidence-level upper limits on the
corresponding cross-sections are obtained. The production of Dark Matter
particles in association with a single top quark is excluded for masses of a
scalar (vector) mediator up to 4.3 (2.3) TeV, assuming $m_\chi$=1 GeV and the
model couplings $\lambda_q$=0.6 and $\lambda_\chi$=0.4 ($a$=0.5 and
$g_\chi$=1). The production of a single vector-like $T$ quark is excluded for
masses below 1.8 TeV assuming a coupling to the top quark $\kappa_T$=0.5 and a
branching ratio for $T\to Zt$ of 25%.
