Inclusive and differential cross-sections for the production of top quarks in association with a photon are measured with proton-proton collision data corresponding to an integrated luminosity of 139 fb$^{−1}$. The data were collected by the ATLAS detector at the LHC during Run 2 between 2015 and 2018 at a centre-of-mass energy of 13 TeV. The measurements are performed in a fiducial volume defined at parton level. Events with exactly one photon, one electron and one muon of opposite sign, and at least two jets, of which at least one is b-tagged, are selected. The fiducial cross-section is measured to be $ {39.6}_{-2.3}^{+2.7} $ fb. Differential cross-sections as functions of several observables are compared with state-of-the-art Monte Carlo simulations and next-to-leading-order theoretical calculations. These include cross-sections as functions of photon kinematic variables, angular variables related to the photon and the leptons, and angular separations between the two leptons in the event. All measurements are in agreement with the predictions from the Standard Model.