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The improved QCD light-cone sum rule (LCSR) provides an effective way to dealwith the heavy-to-light transition form factors (TFFs). Firstly, we adopt theimproved LCSR approach to deal with the $B\to D$ TFF $f^{+}(q^2)$ up to twist-4accuracy. Due to the elimination of the most uncertain twist-3 contribution andthe large suppression of the twist-4 contribution, the obtained LCSR shallprovide us a good platform for testing the $D$-meson leading-twist DA. For thepurpose, we suggest a new model for the $D$-meson leading-twist DA($\phi_{3D}$), whose longitudinal behavior is dominantly determined by aparameter $B$. Moreover, we find its second Gegenbauer moment $a^D_2\sim B$.Varying $B$ within certain region, one can conveniently mimic the $D$-meson DAbehavior suggested in the literature. Inversely, by comparing the estimationswith the experimental data on the $D$-meson involved processes, one can get apossible range for the parameter $B$ and a determined behavior for the$D$-meson DA. Secondly, we discuss the $B\to D$ TFF at the maximum recoilregion and present a detailed comparison of it with the pQCD estimation and theexperimental measurements. Thirdly, by applying the LCSR on $f^{+}(q^2)$, westudy the CKM matrix element $\Vcb$ together with its uncertainties by adoptingtwo types of processes, i.e. the $B^0/\bar{B}^0$-type and the $B^{\pm}$-type.It is noted that a smaller $B \precsim 0.20$ shows a better agreement with theexperimental value on $\Vcb$. For example, for the case of $B=0.00$, we obtain$|V_{cb}|(B^0/\bar{B}^0-{\rm type})=(41.28 {^{+5.68}_{-4.82}}{^{+1.13}_{-1.16}}) \times 10^{-3}$ and $|V_{cb}|(B^{\pm}-{\rm type})=(40.44{^{+5.56}_{-4.72}} {^{+0.98}_{-1.00}}) \times 10^{-3}$, whose first (second)uncertainty comes from the squared average of the mentioned theoretical(experimental) uncertainties.

$V_{cb}$ from the semileptonic decay $B\to D \ell \bar{\nu}_{\ell}$ and the properties of the $D$ meson distribution amplitude