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 $ \sqrt{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 $ {\tilde{t}}_1\to c{\overset{\sim }{\chi}}_1^0 $ or $ {\tilde{t}}_1\to t{\overset{\sim }{\chi}}_1^0 $, where the $ {\overset{\sim }{\chi}}_1^0 $ 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+{E}_T^{\textrm{miss}} $. 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 $ m\left({\tilde{t}}_1\right) $ vs. $ m\left({\overset{\sim }{\chi}}_1^0\right) $ plane and, in addition, limits on the branching ratio of the $ {\tilde{t}}_1\to t{\overset{\sim }{\chi}}_1^0 $ decay as a function of m($ {\tilde{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.[graphic not available: see fulltext]