  We study the impact of the recently computed mixed QCD-electroweak
corrections to the production of $W$ and $Z$ bosons at the LHC on the value of
the $W$ mass extracted from the transverse momentum distribution of charged
leptons from $W$ decays. Using the average lepton transverse momenta in $W$ and
$Z$ decays as simplified observables for the determination of the $W$ mass, we
estimate that mixed QCD-electroweak corrections can shift the extracted value
of the $W$ mass by up to ${\cal O}(20)~{\rm MeV}$, depending on the kinematic
cuts employed to define fiducial cross sections for $Z$ and $W$ production.
Since the target precision of the $W$-mass measurement at the LHC is ${\cal
O}(10)~{\rm MeV}$, our results emphasize the need for fully-differential
computations of mixed QCD-electroweak corrections and a careful analysis of
their potential impact on the determination of the $W$ mass.
