  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
$\sqrt{s}=13~$TeV with the ATLAS experiment, corresponding to an integrated
luminosity of 139$~$fb$^{-1}$. Events are selected with exactly one oppositely
charged electron$-$muon pair and at least one hadronic jet with a transverse
momentum of $p_{\mathrm{T}}>30~$GeV and a pseudorapidity of $|\eta|<4.5$. 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.
