A search for a charged Higgs boson, H$^{±}$, produced in top-quark decays, t → H$^{±}$b, is presented. The search targets H$^{±}$ decays into a bottom and a charm quark, H$^{±}$ → cb. The analysis focuses on a selection enriched in top-quark pair production, where one top quark decays into a leptonically decaying W boson and a bottom quark, and the other top quark decays into a charged Higgs boson and a bottom quark. This topology leads to a lepton-plus-jets final state, characterised by an isolated electron or muon and at least four jets. The search exploits the high multiplicity of jets containing b-hadrons, and deploys a neural network classifier that uses the kinematic differences between the signal and the background. The search uses a dataset of proton-proton collisions collected at a centre-of-mass energy $ \sqrt{s} $ = 13 TeV between 2015 and 2018 with the ATLAS detector at CERN’s Large Hadron Collider, amounting to an integrated luminosity of 139 fb$^{−1}$. Observed (expected) 95% confidence-level upper limits between 0.15% (0.09%) and 0.42% (0.25%) are derived for the product of branching fractions $ \mathcal{B} $(t → H$^{±}$b) × B(H$^{±}$ → cb) for charged Higgs boson masses between 60 and 160 GeV, assuming the SM production of the top-quark pairs.[graphic not available: see fulltext]