A search for flavour-changing neutral interactions involving the top quark, the Higgs boson and an up-type quark q ( q = c, u) is presented. The proton-proton collision data set used, with an integrated luminosity of 139 fb(-1), was collected at root s = 13TeV by the ATLAS experiment at the Large Hadron Collider. Both the decay process t -> qH in tt production and the production process pp. tH, with the Higgs boson decaying into two photons, are investigated. No significant excess is observed and upper limits are set on the t. cH and the t. uH branching ratios of 4.3x10(-4) and 3.8x10(-4), respectively, at the 95% confidence level, while the expected limits in the absence of signal are 4.7x10(-4) and 3.9x10(-4). Combining this search with ATLAS searches in the H. t+ t- and H. b <overline> b final states yields observed (expected) upper limits on the t -> cH branching ratio of 5.8 x 10(-4) (3.0 x 10(-4)) at the 95% confidence level. The corresponding observed (expected) upper limit on the t -> uH branching ratio is 4.0 x 10(-4) (2.4 x 10(-4)).