The results of a search for gluino and squark pair production with the pairs decaying via the lightest charginos into a final state consisting of two $W$ bosons, the lightest neutralinos ($\chi ~10$), and quarks, are presented. The signal is characterised by the presence of a single charged lepton ($e\pm$ or $\mu \pm$) from a $W$ boson decay, jets, and missing transverse momentum. The analysis is performed using 139 fb$\mathrm{<mrow\; data-mjx-texclass="ORD">-1</mrow>}$ of proton-proton collision data taken at a centre-of-mass energy $s=13$ TeV delivered by the Large Hadron Collider and recorded by the ATLAS experiment. No statistically significant excess of events above the Standard Model expectation is found. Limits are set on the direct production of squarks and gluinos in simplified models. Masses of gluino (squark) up to 2.2 TeV (1.4 TeV) are excluded at 95% confidence level for a light $\chi ~10$.