  In a previous paper, we had modified Non-Relativistic QCD as it applies to
quarkonium production by taking into account the effect of perturbative
soft-gluon emission from the colour-octet quarkonium states. We tested the
model by fitting the unknown non-perturbative parameter in the model from
Tevatron data and using that to make parameter-free predictions for $J/\psi$
and $\psi '$ production at the LHC. In this paper, we study $\chi_c$
production: we fit as before the unknown matrix-element using data from
Tevatron. We, then, extend the results of the previous paper for $J/\psi$
production by calculating the effect of $\chi_c$ feed-down to the $J/\psi$
cross-section, which, by comparing with CMS results at $\sqrt{s}=$ 13 TeV, we
demonstrate to be small. We have also computed $\chi_c^1$ and $\chi_c^2$ at
$\sqrt{s}=$7 TeV and find excellent agreement with data from the ATLAS
experiment.
