This paper presents a search for top-squark pair production in final states with a top quark, a charm quark and missing transverse momentum. The data were collected with the ATLAS detector during LHC Run 2 and correspond to an integrated luminosity of 139 fb−1 of proton-proton collisions at a centre-of-mass energy of s = 13 TeV. The analysis is motivated by an extended Minimal Supersymmetric Standard Model featuring a non-minimal flavour violation in the second- and third-generation squark sector. The top squark in this model has two possible decay modes, either t~1→cχ~10 or t~1→tχ~10, where the χ~10 is undetected. The analysis is optimised assuming that both of the decay modes are equally probable, leading to the most likely final state of tc+ETmiss. Good agreement is found between the Standard Model expectation and the data in the search regions. Exclusion limits at 95% CL are obtained in the mt~1 vs. mχ~10 plane and, in addition, limits on the branching ratio of the t~1→tχ~10 decay as a function of m(t~1) are also produced. Top-squark masses of up to 800 GeV are excluded for scenarios with light neutralinos, and top-squark masses up to 600 GeV are excluded in scenarios where the neutralino and the top squark are almost mass degenerate.