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We calculate the $D\to P$ transition form factors within the framework of thelight-cone QCD sum rules (LCSR) with the $D$-meson light-cone distributionamplitudes (LCDAs). The next-to-leading power (NLP) corrections to thevacuum-to-$D$-meson correlation function are considered, and the NLPcorrections from the high-twist $D$-meson LCDAs and the SU(3) breaking effectfrom 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 confirmsthe results from LCSR with pion LCDA. The numerical predictions of the formfactors show that the contribution from two-particle higher-twist contributionsis of great importance and the uncertainties are dominated by the inversemoment of $\phi_D^+(\omega, \mu)$. With the obtained form factors, thepredicted 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 $|V_{cs}|=0.89\,{}^{+0.467}_{-0.234} \big |_{\rmth.}\,{}^{+0.008}_{-0.008} \big |_{\rm exp.}$.

Higher-twist corrections to $D\to\pi,K$ form factors from light-cone sum rules