The production of a $W$ boson in association with a single charm quark is studied using $140\text{ }\text{ }{\mathrm{fb}}^{\ensuremath{-}1}$ of $\sqrt{s}=13\text{ }\text{ }\mathrm{TeV}$ proton-proton collision data collected with the ATLAS detector at the Large Hadron Collider. The charm quark is tagged by the presence of a charmed hadron reconstructed with a secondary-vertex fit. The $W$ boson is reconstructed from the decay to either an electron or a muon and the missing transverse momentum present in the event. The charmed mesons reconstructed are ${D}^{+}\ensuremath{\rightarrow}{K}^{\ensuremath{-}}{\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{+}$ and ${D}^{*+}\ensuremath{\rightarrow}{D}^{0}{\ensuremath{\pi}}^{+}\ensuremath{\rightarrow}({K}^{\ensuremath{-}}{\ensuremath{\pi}}^{+}){\ensuremath{\pi}}^{+}$ and the charge conjugate decays in the fiducial regions where ${p}_{\mathrm{T}}(e,\ensuremath{\mu})>30\text{ }\text{ }\mathrm{GeV}$, $|\ensuremath{\eta}(e,\ensuremath{\mu})|<2.5$, ${p}_{\mathrm{T}}({D}^{(*)})>8\text{ }\text{ }\mathrm{GeV}$, and $|\ensuremath{\eta}({D}^{(*)})|<2.2$. The integrated and normalized differential cross sections as a function of the pseudorapidity of the lepton from the $W$ boson decay, and of the transverse momentum of the charmed hadron, are extracted from the data using a profile likelihood fit. The measured total fiducial cross sections are ${\ensuremath{\sigma}}_{\mathrm{fid}}^{\mathrm{OS}--\mathrm{SS}}({W}^{\ensuremath{-}}+{D}^{+})=50.2\ifmmode\pm\else\textpm\fi{}0.2{(\mathrm{stat})}_{\ensuremath{-}2.3}^{+2.4}(\mathrm{syst})\text{ }\text{ }\mathrm{pb}$, ${\ensuremath{\sigma}}_{\mathrm{fid}}^{\mathrm{OS}--\mathrm{SS}}({W}^{+}+{D}^{\ensuremath{-}})=48.5\ifmmode\pm\else\textpm\fi{}0.2{(\mathrm{stat})}_{\ensuremath{-}2.2}^{+2.3}(\mathrm{syst})\text{ }\text{ }\mathrm{pb}$, ${\ensuremath{\sigma}}_{\mathrm{fid}}^{\mathrm{OS}--\mathrm{SS}}({W}^{\ensuremath{-}}+{D}^{*+})=51.1\ifmmode\pm\else\textpm\fi{}0.4{(\mathrm{stat})}_{\ensuremath{-}1.8}^{+1.9}(\mathrm{syst})\text{ }\text{ }\mathrm{pb}$, and ${\ensuremath{\sigma}}_{\mathrm{fid}}^{\mathrm{OS}--\mathrm{SS}}({W}^{+}+{D}^{*\ensuremath{-}})=50.0\ifmmode\pm\else\textpm\fi{}0.4{(\mathrm{stat})}_{\ensuremath{-}1.8}^{+1.9}(\mathrm{syst})\text{ }\text{ }\mathrm{pb}$. Results are compared with the predictions of next-to-leading-order quantum chromodynamics calculations performed using state-of-the-art parton distribution functions. Additionally, the ratio of charm to anticharm production cross sections is studied to probe the $s\text{\ensuremath{-}}\overline{s}$ quark asymmetry. The ratio is found to be ${R}_{c}^{\ifmmode\pm\else\textpm\fi{}}=0.971\ifmmode\pm\else\textpm\fi{}0.006(\mathrm{stat})\ifmmode\pm\else\textpm\fi{}0.011(\mathrm{syst})$. The ratio and cross-section measurements are consistent with the predictions obtained with parton distribution function sets that have a symmetric $s\text{\ensuremath{-}}\overline{s}$ sea, indicating that any $s\text{\ensuremath{-}}\overline{s}$ asymmetry in the Bjorken-$x$ region relevant for this measurement is small.