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We study the quasi-two-body $D\to TP$ decays and the three-body $D$ decaysproceeding through intermediate tensor resonances, where $T$ and $P$ denotetensor and pseudoscalar mesons, respectively. We employ $D\to T$ transitionform factors based upon light-cone sum rules and the covariant light-frontquark model to evaluate the decay rates, with the former giving a betteragreement with current data. Though the tree amplitudes with the emitted mesonbeing a tensor meson vanish under factorization approximation, contributionsproportional to the tensor decay constant $f_T$ can be produced from vertex andhard spectator-scattering corrections. We also investigate the finite-widtheffects of the tensor mesons and find that, contrary to three-body $B$ decays,the tensor-mediated $D$ decays are more seriously affected and the narrow widthapproximation has to be corrected. More experimental data are required in orderto extract information topological amplitudes associated with quasi-two-body$D\to TP$ decays. Among the data, the $D^+\to f_2\pi^+$ and $D^+\to \barK_2^{*0}\pi^+$ branching fractions are not self consistent and furtherclarification is called for.

Two- and three-body hadronic decays of charmed mesons involving a tensor meson