  We propose a method to help control cutoff effects in the short-distance
contribution to integrated correlation functions, such as the hadronic vacuum
polarization (HVP), using the corresponding screening correlators computed at
finite temperature. The strategy is investigated with Wilson fermions at
leading order, which reveals a logarithmically-enhanced lattice artifact in the
short-distance contribution, whose coefficient is determined at this order. We
then perform a numerical study with $N_\mathrm{f}=2$ O($a$)-improved Wilson
fermions and a temperature $T\approx250~\mathrm{MeV}$, with lattice spacings
down to $a\approx0.03~\mathrm{fm}$, which suggests good control can be achieved
on the short-distance contribution to the HVP and the Adler function at large
virtuality. Finally, we put forward a scheme to compute the complete HVP
function at arbitrarily large virtualities using a step-scaling in the
temperature.
