  We consider the exclusive pseudoscalar heavy-quarkonium (eta_{b,c})
production in association with a photon at future lepton colliders where the
collider energies of O(10^2) GeV are far greater than the quarkonium mass. At
these energies, the logarithm of mass to collision energy becomes increasingly
large hence its resummation becomes particularly important. By making use of
the light-cone-distribution factorization formula, we resum the logarithms up
to next-to-leading-logarithmic accuracy (NLL) that corresponds to order-alpha_s
accuracy. We combine the resummed result with a known fixed-order result at
next-to-leading order (NLO) such that both resummed-logarithmic terms and
non-logarithmic terms are included at the same order in alpha_s. This allowed
us to provide reliable predictions at accuracies of order alpha_s ranging from
relatively low energies near quarkonium mass to the collider energies of
O(10^2) GeV. We also include the leading relativistic corrections resummed at
leading-logarithmic accuracy. Our prediction at the Belle energy is comparable
with fixed-order predictions in literatures while it shows a large deviation
from a recent Belle's upper limit by about 4 sigma. Finally, we make
predictions for the energies of future Z and Higgs factories.
