Inclusive cross-sections for top-quark pair production in association with charm quarks are measured with proton–proton collision data at a center-of-mass energy of 13 TeV corresponding to an integrated luminosity of 140 fb−1, collected with the ATLAS experiment at the LHC between 2015 and 2018. The measurements are performed by requiring one or two charged leptons (electrons and muons), two b-tagged jets, and at least one additional jet in the final state. A custom flavor-tagging algorithm is employed for the simultaneous identification of b-jets and c-jets. In a fiducial phase space that replicates the acceptance of the ATLAS detector, the cross-sections for tt¯+≥2c and tt¯+1c production are measured to be 1.28−0.24+0.27pb and 6.4−0.9+1.0pb, respectively. The measurements are primarily limited by uncertainties in the modeling of inclusive tt¯ and tt¯+bb¯ production, in the calibration of the flavor-tagging algorithm, and by data statistics. Cross-section predictions from various tt¯ simulations are largely consistent with the measured cross-section values, though all underpredict the observed values by 0.5 to 2.0 standard deviations. In a phase-space volume without requirements on the tt¯ decay products and the jet multiplicity, the cross-section ratios of tt¯+≥2c and tt¯+1c to total tt¯+jets production are determined to be (1.23±0.25)% and (8.8±1.3)%.