  Inspired by a series of unexpected measurements of semileptonic decays
mediated via $b\rightarrow c $ charged current interactions, we explore
semileptonic $B_c$ decays to the four lightest $P$-wave charmonium states,
$\chi_{c0}, \chi_{c1}, \chi_{c2}, h_c$, by the recently developed improved
perturbative QCD formalism, in which the charm quark mass effect is included
both in the Sudakov factor and the hard kernels. We first directly evaluate the
concerned transition form factors with vector and axial-vector currents in the
region of small momentum transfer squared, and then recast them to the full
kinematical region by adopting the exponential parametrization. The obtained
form factors are used to evaluate the semileptonic decay branching ratios,
which can reach the order of $10^{-3}$, letting the corresponding measurement
appear feasible. For a better analysis, a comparison of our results with the
predictions of other models is provided. We also present the ratios between the
tau and light lepton branching ratios and the polarization contributions in the
relevant processes, which still need experimental tests in the ongoing and
forthcoming experiments. Any significant deviations from the Standard Model
results may provide some hints of new physics effects.
