Several extensions of the Standard Model predict the production of dark matter particles at the LHC. A search for dark matter particles produced in association with a dark Higgs boson decaying into W$^{+}$W$^{−}$ in the $ {\ell}^{\pm}\nu q{\overline{q}}^{\prime }, $ final states with ℓ = e, μ is presented. This analysis uses 139 fb$^{−1}$ of pp collisions recorded by the ATLAS detector at a centre-of-mass energy of 13 TeV. The W$^{±}$ → $ q\overline{q^{\prime }} $ decays are reconstructed from pairs of calorimeter-measured jets or from track-assisted reclustered jets, a technique aimed at resolving the dense topology from a pair of boosted quarks using jets in the calorimeter and tracking information. The observed data are found to agree with Standard Model predictions. Scenarios with dark Higgs boson masses ranging between 140 and 390 GeV are excluded.[graphic not available: see fulltext]