  We investigated the form factors of the $B_{c}$ meson into $P$-wave orbitally
excited charmonium using the nonrelativistic QCD effective theory. Through the
analytic computation, the next-to-leading order relativistic corrections to the
form factors were obtained, and the asymptotic expressions were studied in the
infinite bottom quark mass limit. Employing the general form factors, we
discussed the exclusive decays of the $B_{c}$ meson into $P$-wave orbitally
excited charmonium and a light meson. We found that the relativistic
corrections lead to a large correction for the form factors, which makes the
branching ratios of the decay channels $\mathcal{B}(B_{c} ^{\pm }\to \chi
_{cJ}(h_c) +\pi ^{\pm }(K^{\pm }))$ larger. These results are useful for the
phenomenological analysis of the $B_{c}$ meson decays into $P$-wave charmonium,
which shall be tested in the LHCb experiments.
