  We compute leading-power fragmentation corrections to $J/\psi$
photoproduction at DESY HERA, making use of the nonrelativistic QCD
factorization approach. Our calculations include parton production cross
sections through order $\alpha_s^3$, fragmentation functions though order
$\alpha_s^2$, and leading logarithms of the transverse momentum divided by the
charm-quark mass to all orders in $\alpha_s$. We find that the leading-power
fragmentation corrections, beyond those that are included through
next-to-leading order in $\alpha_s$, are small relative to the fixed-order
contributions through next-to-leading order in $\alpha_s$. Consequently, an
important discrepancy remains between the experimental measurements of the
$J/\psi$ photoproduction cross section and predictions that make use of
nonrelativistic-QCD long-distance matrix elements that are extracted from the
$J/\psi$ hadroproduction cross-section and polarization data.
