Inclusive and differential cross-sections are measured at particle level for the associated production of a top quark pair and a photon ($ t\overline{t}\gamma $). The analysis is performed using an integrated luminosity of 140 fb$^{−1}$ of proton-proton collisions at a centre-of-mass energy of 13 TeV collected by the ATLAS detector. The measurements are performed in the single-lepton and dilepton top quark pair decay channels focusing on $ t\overline{t}\gamma $ topologies where the photon is radiated from an initial-state parton or one of the top quarks. The absolute and normalised differential cross-sections are measured for several variables characterising the photon, lepton and jet kinematics as well as the angular separation between those objects. The observables are found to be in good agreement with the Monte Carlo predictions. The photon transverse momentum differential distribution is used to set limits on effective field theory parameters related to the electroweak dipole moments of the top quark. The combined limits using the photon and the Z boson transverse momentum measured in $ t\overline{t} $ production in associations with a Z boson are also set.[graphic not available: see fulltext]