Fiducial and differential measurements of $W+W-$ production in events with at least one hadronic jet are presented. These cross-section measurements are sensitive to the properties of electroweak-boson self-interactions and provide a test of perturbative quantum chromodynamics and the electroweak theory. The analysis is performed using proton$-$proton collision data collected at $s=13$TeV with the ATLAS experiment, corresponding to an integrated luminosity of 139$$fb$\mathrm{<mrow\; data-mjx-texclass="ORD">-1</mrow>}$. Events are selected with exactly one oppositely charged electron$-$muon pair and at least one hadronic jet with a transverse momentum of $pT>30$GeV and a pseudorapidity of . After subtracting the background contributions and correcting for detector effects, the jet-inclusive $W+W-+\ge 1$jet fiducial cross-section and $W+W-+$ jets differential cross-sections with respect to several kinematic variables are measured, thus probing a previously unexplored event topology at the LHC. These measurements include leptonic quantities, such as the lepton transverse momenta and the transverse mass of the $W+W-$ system, as well as jet-related observables such as the leading jet transverse momentum and the jet multiplicity. Limits on anomalous triple-gauge-boson couplings are obtained in a phase space where interference between the Standard Model amplitude and the anomalous amplitude is enhanced.