Abstract: A search for flavour-changing neutral current (FCNC) 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 -> gamma u of 2.8 x 10(-5) (6.1 x 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.coupling of 40 fb (33 fb) and on the branching ratio for t -> gamma c of 22 x 10(-5) (18 x 10(-5)).