  We use a new approach to color evaporation model (CEM) for quarkonium
production. The production of $c\bar c$ pairs is performed within
$k_T$-factorization approach using different unintegrated gluon distribution
functions (UGDF) from the literature. We include all recent improvements to
color evaporation model. We cannot describe simultaneously mid and forward
rapidity data measured at the LHC when using the KMR UGDF with the same
normalization parameter. Furthermore we get somewhat too hard distribution in
$J/\psi$ transverse momentum. Correcting the standard KMR distributions for
saturation effects at small values of $x$ improves $J/\psi$ rapidity
distributions, while taking correction for emissions hidden in the KMR UGDF in
the evaluation of $x$ values improves a bit $J/\psi$ transverse momentum
distributions. The modifications improve also description of the LHCb $D$-meson
data.
