  We study the quasi-two-body $D\to TP$ decays and the three-body $D$ decays
proceeding through intermediate tensor resonances, where $T$ and $P$ denote
tensor and pseudoscalar mesons, respectively. We employ $D\to T$ transition
form factors based upon light-cone sum rules and the covariant light-front
quark model to evaluate the decay rates, with the former giving a better
agreement with current data. Though the tree amplitudes with the emitted meson
being a tensor meson vanish under factorization approximation, contributions
proportional to the tensor decay constant $f_T$ can be produced from vertex and
hard spectator-scattering corrections. We also investigate the finite-width
effects 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 width
approximation has to be corrected. More experimental data are required in order
to extract information topological amplitudes associated with quasi-two-body
$D\to TP$ decays. Among the data, the $D^+\to f_2\pi^+$ and $D^+\to \bar
K_2^{*0}\pi^+$ branching fractions are not self consistent and further
clarification is called for.
