  In this paper, we study the leading-twist distribution amplitude (DA) of the
heavy pseudoscalars (HPs), such as $\eta_c$, $\eta_b$ and $B_c$, within the QCD
theory in the background fields. New sum rules up to dimension-six condensates
for both the HP decay constants and their leading-twist DA moments are
presented. From the sum rules for the HP decay constants, we obtain $f_{\eta_c}
= 453 \pm 4 \textrm{MeV}$, $f_{B_c} = 498 \pm 14 \textrm{MeV}$, and $f_{\eta_b}
= 811 \pm 34 \textrm{MeV}$. Basing on the sum rules for the HPs' leading-twist
DA moments, we construct a new model for the $\eta_c$, $\eta_b$ and $B_c$
leading-twist DAs. Our present HP DA model can also be adaptable for the light
pseudo-scalar DAs, such as the pion and kaon DAs. Thus, it shall be applicable
for a wide range of QCD exclusive processes. As an application, we apply the
$\eta_c$ leading-twist DA to calculate the $B_c \to \eta_c$ transition form
factor $f_+^{B_c \to \eta_c}(q^2)$. At the maximum recoil region, we obtain
$f_+^{B_c \to \eta_c}(0) = 0.612^{+0.053}_{-0.052}$. After further
extrapolating the TFF $f_+^{B_c \to \eta_c}(q^2)$ to its allowable $q^2$
region, we predict the branching ratio for the semi-leptonic decay $B_c \to
\eta_c l \nu$. We obtain ${\cal B}(B_c \to \eta_c l
\nu)=\left(7.70^{+1.65}_{-1.48}\right) \times 10^{-3}$ for massless leptons,
which is consistent with the LCSRs estimation obtained in the literature.
