A search for a light charged Higgs boson produced in decays of the top quark, $t \rightarrow H^{\pm } b$ with $H^{\pm } \rightarrow cs$, is presented. This search targets the production of top-quark pairs $t\bar{t} \rightarrow Wb H^{\pm } b$, with $W \rightarrow \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 $140\hbox { fb}^{-1}$ of $pp$ collisions at $\sqrt{s} = 13\hbox { TeV}$ recorded with the ATLAS detector at the LHC between 2015 and 2018. Observed (expected) 95% confidence-level upper limits on the branching fraction $\mathscr {B}(t\rightarrow H^{\pm } b)$, assuming $\mathscr {B}(t\rightarrow Wb) + \mathscr {B}(t \rightarrow H^{\pm } (\rightarrow cs)b)=1.0$, 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.