This dissertation presents a search for flavour-changing neutral-current (FCNC) interactions between the top quark, the Higgs boson and either the up or the charm quark. The search is performed using a dataset corresponding to an integrated luminosity of 140 1/fb, recorded with the ATLAS detector at the Large Hadron Collider at a centre-of-mass energy of 13 TeV. The anomalous FCNC couplings are parametrised using an effective field theory. The interactions are analysed in two distinct modes: the production of a top quark-antiquark pair, with one of the top quarks performing an FCNC decay into a Higgs boson and a different up-type quark, and the associated production of a single top quark and a Higgs boson via the new FCNC vertex. Decay modes of the Higgs boson to two W bosons, two Z bosons or two tau leptons are considered with final states containing either exactly two leptons of the same charge, or three leptons, exactly two of which possess an identical charge. No deviations from SM expectations are observed in the statistical analysis. Therefore, 95 % confidence level upper exclusionlimits are set on the FCNC top-quark decay branching ratios, amounting to observed (expected) limits of B(t Hu) < 0.028 (0.030) % and B(t Hc) < 0.033 (0.038) %.