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In the wake of measurements on $B_c^+ \to J/\psi K^+$, $B_c^+ \to J/\psi\pi^+\pi^-\pi^+$, and $B_c^+ \to J/\psi K^+ K^-\pi^+$ at Large Hadron Colliderexperiments, we propose to study the decays $B_c^+ \to J/\psi M^+$comprehensively, with $M$ being the light charged pseudoscalar ($P$), vector($V$), scalar ($S$), axial-vector ($A$), and tensor ($T$) mesons, within theimproved perturbative QCD (iPQCD) formalism at leading order in the StandardModel. The theoretical predictions for experimental observables such asbranching fractions, relative ratios, and longitudinal polarization fractionsin the iPQCD formalism await near future examinations relying on the upgradedLarge Hadron Collider, even the forthcoming Circular Electron-PositronCollider. We emphasize that the investigations on thefactorizable-emission-suppressed or -forbidden decays like $B_c^+ \to J/\psiS^+$, $B_c^+ \to J/\psi A^+_{1^1\!P_1}$, and $B_c^+ \to J/\psi T^+$, should godefinitely beyond naive factorization to explore the rich dynamics, whichcould, in turn, further help understand the QCD nature of $B_c$ meson, as wellas that of related hadrons. The future confirmations on those predictions aboutthe relative ratios between the branching fractions of $B_c^+ \to J/\psib_1(1235)^+ (a_0(980)^+, a_0(1450)^+, a_2(1320)^+)$ and $B_c^+ \to J/\psi\pi^+$ could further examine the reliability of this iPQCD formalism. Becauseof containing only tree-level $\bar b \to \bar c$ transitions, the {\it CP}asymmetries in the $B_c^+ \to J/\psi M^+$ decays exhibit naturally zero.

$B_c$-meson decays into $J/\psi$ plus a light meson in the improved perturbative QCD formalism