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We calculate the $D\to P$ transition form factors within the framework of the light-cone QCD sum rules (LCSR) with the $D$-meson light-cone distribution amplitudes (LCDAs). The next-to-leading power (NLP) corrections to the vacuum-to-$D$-meson correlation function are considered, and the NLP corrections from the high-twist $D$-meson LCDAs and the SU(3) breaking effect from strange quark mass are investigated. Adopting the exponential model of the $D$-meson LCDAs,the predicted SU(3) flavor symmetry breaking effects are $R_{SU(3)}^{+,0}=1.12$ and $R_{SU(3)}^{T}=1.39$, respectively, which confirms the results from LCSR with pion LCDA. The numerical predictions of the form factors show that the contribution from two-particle higher-twist contributions is of great importance and the uncertainties are dominated by the inverse moment of $\phi_D^+(\omega, \mu)$. With the obtained form factors, the predicted Cabibbo-Kobayashi-Maskawa (CKM) matrix elements are $|V_{cd}|=0.151\,{}^{+0.091}_{-0.043} \big |_{\rm th.}\,{}^{+0.017}_{-0.02} \big |_{\rm exp.}$ and $|V_{cs}|=0.89\,{}^{+0.467}_{-0.234} \big |_{\rm th.}\,{}^{+0.008}_{-0.008} \big |_{\rm exp.}$.