  The $B\to D$ transition form factor (TFF) $f^{B\to D}_+(q^2)$ is determined
mainly by the $D$-meson leading-twist distribution amplitude (DA) ,
$\phi_{2;D}$, if the proper chiral current correlation function is adopted
within the light-cone QCD sum rules. It is therefore significant to make a
comprehensive study of DA $\phi_{2;D}$ and its impact on $f^{B\to D}_+(q^2)$.
In this paper, we calculate the moments of $\phi_{2;D}$ with the QCD sum rules
under the framework of the background field theory. New sum rules for the
leading-twist DA moments $\left<\xi^n\right>_D$ up to fourth order and up to
dimension-six condensates are presented. At the scale $\mu = 2 \rm GeV$, the
values of the first four moments are: $\left<\xi^1\right>_D =
-0.418^{+0.021}_{-0.022}$, $\left<\xi^2\right>_D = 0.289^{+0.023}_{-0.022}$,
$\left<\xi^3\right>_D = -0.178 \pm 0.010$ and $\left<\xi^4\right>_D =
0.142^{+0.013}_{-0.012}$. Basing on the values of
$\left<\xi^n\right>_D(n=1,2,3,4)$, a better model of $\phi_{2;D}$ is
constructed. Applying this model for the TFF $f^{B\to D}_+(q^2)$ under the
light cone sum rules, we obtain $f^{B\to D}_+(0) = 0.673^{+0.038}_{-0.041}$ and
$f^{B\to D}_+(q^2_{\rm max}) = 1.124^{+0.053}_{-0.058}$. The uncertainty of
$f^{B\to D}_+(q^2)$ from $\phi_{2;D}$ is estimated and we find its impact
should be taken into account, especially in low and central energy region. The
branching ratio $\mathcal{B}(B\to Dl\bar{\nu}_l)$ is calculated, which is
consistent with experimental data.
