Cross-section measurements of top-quark pair production in association with a hard photon at 13 TeV with the ATLAS detector

25 years after the top quark's discovery, the Large Hadron Collider at CERN produces proton-proton collision data on unprecedented scales at unprecedented energies – and has heralded an era of top-quark precision measurements. The production of a top-quark pair in association with a photon ($t\bar{t}\gamma$) gives access to the electromagnetic top-photon coupling, one of the fundamental properties of the top quark. Various extensions of the Standard Model predict modifications of the coupling strength or structure, and deviations from the Standard Model prediction of the $t\bar{t}\gamma$ production cross-section would indicate new physics. With enough statistics available from the Large Hadron Collider, the electron-muon channel has gained particular interest due to its high signal purity and precise available theory predictions. This thesis presents results with the full Run 2 dataset collected with the ATLAS detector in proton-proton collisions at the Large Hadron Collider between 2015 and 2018 at 13 TeV centre-of-mass energy, corresponding to an integrated luminosity of 139 fb$^{-1}$. In order to compare the results to fixed-order calculations that include non-doubly-resonant diagrams, a combined measurement of $t\bar{t}\gamma + tW\gamma$ is performed. The focus is placed on a measurement of the fiducial inclusive cross-section in the electron-muon channel, where exactly one photon, one electron and one muon of opposite charge sign, and at least two jets, one of which must be $b$-tagged, are selected. Furthermore, the ATLAS data is unfolded to parton level and measurements of differential cross-sections as functions of several observables are presented. Both fiducial inclusive and differential results are compared to state-of-the-art fixed-order calculations at next-to-leading order in QCD. An additional focus of the thesis is placed on studies to use machine-learning techniques, in particular deep neural networks, for the identification of prompt photons.