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We study the quasi-two-body $D\to SP$ decays and the three-body $D$ decays proceeding through intermediate scalar resonances, where $S$ and $P$ denote scalar and pseudoscalar mesons, respectively. Our main results are: (i) Certain external and internal $W$-emission diagrams with the emitted meson being a scalar meson are na{ï}vely expected to vanish, but they actually receive contributions from vertex and hard spectator-scattering corrections beyond the factorization approximation. (ii) For light scalars with masses below or close to 1~GeV, it is more sensible to study three-body decays directly and compare with experiment as the two-body branching fractions are either unavailable or subject to large finite-width effects of the scalar meson. (iii) We consider the two-quark (scheme I) and four-quark (scheme II) descriptions of the light scalar mesons, and find the latter generally in better agreement with experiment. This is in line with recent BESIII measurements of semileptonic charm decays that prefer the tetraquark description of light scalars produced in charmed meson decays. (iv) The topological amplitude approach fails here as the $D\to SP$ decay branching fractions cannot be reliably inferred from the measurements of three-body decays, mainly because the decay rates cannot be factorized into the topological amplitude squared and the phase space factor. (v) The predicted rates for $D^0\to f_0 P, a_0 P$ are generally smaller than experimental data by one order of magnitude, presumably implying the significance of $W$-exchange amplitudes. (vi) The $W$-annihilation amplitude is found to be very sizable in the $SP$ sector with $|A/T|_{SP}\sim 1/2$, contrary to its suppression in the $PP$ sector with $|A/T|_{PP}\sim 0.18$.