  We improve the theoretical predictions for the decays of the Higgs boson to
an $S$-wave vector quarkonium plus a photon by calculating the relativistic
correction of order $v^2$, where $v$ is the heavy-quark velocity in the
quarkonium rest frame. Our numerical results are given for the $J/\psi$ and
$\Upsilon(nS)$ channels, with $n=1,2,3$. The numerical results include a
previously calculated correction of order $\alpha_s$ and summations, to all
orders in $\alpha_s$, of leading logarithms of $m_H^2/m_Q^2$, where $m_H$ is
the Higgs-boson mass and $m_Q$ is the heavy-quark mass. These QCD corrections
apply to the contribution of leading order in $v$ and to part of the
order-$v^2$ correction. For the remainder of the order-$v^2$ correction, we sum
leading logarithms of $m_H/m_Q$ through order $\alpha_s^2$. These refinements
reduce the theoretical uncertainties in the direct-production amplitudes for $H
\to J/\psi+\gamma$ and $H \to \Upsilon(1S)+\gamma$ by approximately a factor of
3 and open the door to improved determinations at the LHC of the Higgs-boson
Yukawa couplings to the charm and bottom quarks.
