A bstract 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 $$ \sqrt{s} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msqrt> <mml:mi>s</mml:mi> </mml:msqrt> </mml:math> = 13 TeV by the ATLAS experiment at the Large Hadron Collider. Both the decay process t → qH in $$ t\overline{t} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>t</mml:mi> <mml:mover> <mml:mi>t</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> </mml:math> 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 . 3 × 10 − 4 and 3 . 8 × 10 − 4 , respectively, at the 95% confidence level, while the expected limits in the absence of signal are 4 . 7 × 10 − 4 and 3 . 9 × 10 − 4 . Combining this search with ATLAS searches in the H → τ + τ − and H → $$ b\overline{b} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>b</mml:mi> <mml:mover> <mml:mi>b</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> </mml:math> final states yields observed (expected) upper limits on the t → cH branching ratio of 5 . 8 × 10 − 4 (3 . 0 × 10 − 4 ) at the 95% confidence level. The corresponding observed (expected) upper limit on the t → uH branching ratio is 4 . 0 × 10 − 4 (2 . 4 × 10 − 4 ).