  A search is presented for the production of a single top quark via
left-handed flavour-changing neutral-current (FCNC) interactions of a top
quark, a gluon and an up or charm quark. Two production processes are
considered: $u+g\rightarrow t$ and $c+g\rightarrow t$. The analysis is based on
proton-proton collision data taken at a centre-of-mass energy of 13$\,$TeV with
the ATLAS detector at the LHC. The data set corresponds to an integrated
luminosity of 139 fb$^{-1}$. Events with exactly one electron or muon, exactly
one $b$-tagged jet and missing transverse momentum are selected, resembling the
decay products of a singly produced top quark. Neural networks based on
kinematic variables differentiate between events from the two signal processes
and events from background processes. The measured data are consistent with the
background-only hypothesis, and limits are set on the production cross-sections
of the signal processes: $\sigma(u+g\rightarrow
t)\times\mathcal{B}(t\rightarrow Wb)\times\mathcal{B}(W\rightarrow
\ell\nu)<3.0\,$pb and $\sigma(c+g\rightarrow t)\times\mathcal{B}(t\rightarrow
Wb)\times\mathcal{B}(W\rightarrow \ell\nu)<4.7\,$pb at the 95% confidence
level, with $\mathcal{B}(W\rightarrow \ell\nu)=0.325$ being the sum of
branching ratios of all three leptonic decay modes of the $W$ boson. Based on
the framework of an effective field theory, the cross-section limits are
translated into limits on the strengths of the $tug$ and $tcg$ couplings
occurring in the theory: $|C^{\,ut}_{uG}|/\Lambda^2 < 0.057\,$TeV$^{-2}$ and
$|C^{\,ct}_{uG}|/\Lambda^2 < 0.14\,$TeV$^{-2}$. These bounds correspond to
limits on the branching ratios of FCNC-induced top-quark decays:
$\mathcal{B}(t\rightarrow u+g)< 0.61\times 10^{-4}$ and
$\mathcal{B}(t\rightarrow c+g)< 3.7\times 10^{-4}$.
