  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 corresponds to an
integrated luminosity of 139fb$^{-1}$ 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 $\tilde{t}_1 \rightarrow
c\tilde{\chi}_1^0$ or $\tilde{t}_1\rightarrow t\tilde{\chi}_1^0$, where the
$\tilde{\chi}_1^0$ is undetected. The analysis is optimised assuming that both
of the decay modes are equally probable, leading to the most likely final state
of $tc + E_{\text{T}}^{\text{miss}}$. 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 $m(\tilde{t}_1)$ vs $m(\tilde{\chi}_1^0)$ plane
and, in addition, limits on the branching ratio of the $\tilde{t}_1\rightarrow
t\tilde{\chi}_1^0$ 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.
