A search for a light charged Higgs boson produced in decays of the top quark, $t \to H^\pm b$ with $H^\pm \to cs$, is presented.This search targets the production of top-quark pairs $t\bar{t} \to Wb H^\pm b$, with $W \to \ell\nu$ ($\ell = e, \mu$),resulting in a lepton-plus-jets final state characterised by an isolated electron or muon and at least four jets.The search exploits $b$-quark and $c$-quark identification techniques as well as multivariate methods to suppress the dominant $t\bar{t}$ background.The data analysed correspond to $140\,\text{fb}^{-1}$ of $pp$ collisions at $\sqrt{s} = 13\,\text{TeV}$ recorded with the ATLAS detector at the LHC between 2015 and 2018.Observed (expected) $95\,$% confidence-level upper limits on the branching fraction $\mathscr{B}(t\to H^\pm b)$, assuming $\mathscr{B}(t\to Wb) + \mathscr{B}(t \to H^\pm (\to cs)b)=1.0$,are set between $0.066\,$% ($0.077\,$%) and $3.6\,$% ($2.3\,$%) for a charged Higgs boson with a mass between 60 and $168\,$GeV.