  In this work, we present a thorough analysis of $\eta_c(1S,2S)$ quarkonia
hadroproduction in $k_\perp$-factorisation in the framework of the light-front
potential approach for the quarkonium wave function. The off-shell matrix
elements for the $g^* g^* \eta_c(1S,2S)$ vertices are derived. We discuss the
importance of taking into account the gluon virtualities. %We discuss the role
of relativistic corrections, the Melosh spin-rotation and theoretical
uncertainties by comparing our results with the standard NRQCD approach. We
present the transverse momentum distributions of $\eta_c$ for several models of
the unintegrated gluon distributions. Our calculations are performed for four
distinct parameterisations for the $c\bar c$ interaction potential consistent
with the meson spectra. We compare our results for $\eta_c(1S)$ to measurements
by the LHCb collaboration and present predictions for $\eta_c(2S)$ production.
