  We compute fragmentation corrections to hadroproduction of the quarkonium
states $J/\psi$, $\chi_{cJ}$, and $\psi(2S)$ at leading power in $m_c^2/p_T^2$,
where $m_c$ is the charm-quark mass and $p_T$ is the quarkonium transverse
momentum. The computation is carried out in the framework of nonrelativistic
QCD. We include corrections to the parton-production cross sections through
next-to-leading order in the strong coupling $\alpha_s$ and corrections to the
fragmentation functions through second order in $\alpha_s$. We also sum leading
logarithms of $p_T^2/m_c^2$ to all orders in perturbation theory. We find that,
when we combine these leading-power fragmentation corrections with fixed-order
calculations through next-to-leading order in $\alpha_s$, we are able to obtain
good fits for $p_T\geq 10$ GeV to hadroproduction cross sections that were
measured at the Tevatron and the LHC. Using values for the nonperturbative
long-distance matrix elements that we extract from the cross-section fits, we
make predictions for the polarizations of the quarkonium states. We obtain good
agreement with measurements of the polarizations, with the exception of the CDF
Run II measurement of the prompt $J/\psi$ polarization, for which the agreement
is only fair. In the predictions for the prompt-$J/\psi$ cross sections and
polarizations, we take into account feeddown from the $\chi_{cJ}$ and
$\psi(2S)$ states.
