  We present results for the short-distance window observable of the hadronic
vacuum polarization contribution to the muon $g-2$, computed via the
time-momentum representation (TMR) in lattice QCD. A key novelty of our
calculation is the reduction of discretization effects by a suitable
subtraction applied to the TMR kernel function, which cancels the leading
$x_0^4$-behaviour at short distances. To compensate for the subtraction, one
must substitute a term that can be reliably computed in perturbative QCD. We
apply this strategy to our data for the vector current collected on ensembles
generated with $2+1$ flavours of O($a$)-improved Wilson quarks at six values of
the lattice spacing and pion masses in the range $130-420\,$MeV. Our estimate
at the physical point contains a full error budget and reads $(a_\mu^{\rm
hvp})^{\rm SD}=68.85(14)_{\rm stat}\,(42)_{\rm syst}\cdot10^{-10}$, which
corresponds to a relative precision of 0.7\%. We discuss the implications of
our result for the observed tensions between lattice and data-driven
evaluations of the hadronic vacuum polarization.
