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 $\mathcal{B}(t\rightarrow c+g)< 3.7\times 10^{-4}$.