  The hadroproduction of a $W$ boson in association with a charm quark at the
Large Hadron Collider is at the centre of current investigations due to its
potential to probe the strangeness content of the proton. In this paper we
present an implementation of the $W+c$ production process in the PowHel event
generator matched to the PYTHIA8 parton shower approach, allowing to obtain
predictions for differential cross-sections with NLO QCD accuracy matched to
the accuracy of the Shower Monte Carlo event generator. Effects of non-diagonal
CKM matrix elements, finite charm quark mass and off-shell $W$ decays including
spin correlations are taken into account. We investigate the production of a
leptonically decaying $W$ boson in association with either a charmed meson
($W^\pm~+~D^{*\mp}$) or a charmed jet ($W^\pm~+~j_{c}$) and compare our
predictions with particle-level measurements by the ATLAS and CMS
collaborations at $\sqrt{s} =$ 7 and 13 TeV. Considering the level of agreement
between theory predictions and experimental data in the light of present
theoretical and experimental uncertainties, our results do not point to the
need of extensive modifications of the strange and antistrange distribution
functions in the NLO PDF fits that we used, although collider $W + c$
production data have not been included yet in these fits.
