CERN-LHC, ATLAS. A search for flavour-changing neutral current events via the coupling of a top quark, a photon, and an up or charm quark is presented using 81$fb^{−1}$ of proton-proton collision data taken at a centre-of-mass energy of 13 TeV with the ATLAS detector at the LHC. Events with a photon, an electron or muon, a $b$-tagged jet, and missing transverse momentum are selected. A neural network based on kinematic variables differentiates between events from signal and background processes. The data are consistent with the background-only hypothesis, and limits are set on the strength of the $tq\gamma$ coupling in an effective field theory. These are also interpreted as 95 % CL upper limits on the cross section for FCNC $t\gamma$ production via a left-handed (right-handed) $tu\gamma$ coupling of 36 fb (78 fb) and on the branching ratio for $t \rightarrow \gamma u$ of $2.8 \times 10^{−5}$ ($6.1 \times 10^{−5}$). In addition, they are interpreted as 95 % CL upper limits on the cross section for FCNC $t\gamma$ production via a left-handed (right-handed) $tc\gamma$ coupling of 40 fb (33 fb) and on the branching ratio for $t \rightarrow \gamma c$ of $22 \times 10^{−5}$ ($18 \times 10^{−5}$).