  In this paper, moments of $a_0(980)$-meson twist-2 light-cone distribution
amplitudes were deeply researched by using QCD sum rules approach within
background field theory. Up to 9th-order accuracy, we present
$\langle\xi_{2;a_0}^n\rangle|_{\mu_0}$ at the initial scale $\mu_0 = 1~{\rm
GeV}$, i.e. $\langle\xi^1_{2;a_0}\rangle|_{\mu_0} = -0.307(43)$,
$\langle\xi^3_{2;a_0}\rangle|_{\mu_0} = -0.181(34)$,
$\langle\xi^5_{2;a_0}\rangle|_{\mu_0} = -0.078(28)$,
$\langle\xi^7_{2;a_0}\rangle|_{\mu_0} = -0.049(26)$,
$\langle\xi^9_{2;a_0}\rangle|_{\mu_0} = -0.036(24)$, respectively. An improved
light-cone harmonic oscillator model for $a_0(980)$-meson twist-2 light-cone
distribution amplitudes is adopted, where its parameters are fixed by using the
least squares method based on the $\langle\xi_{2;a_0}^n\rangle|_{\mu_0}$, and
their goodness of fit reach to $95.4\%$. Then, we calculate the $D\to a_0(980)$
transition form factors within the light-cone sum rules approach, and at
largest recoil point, we obtain $f_+^{D\to a_0}(0) = 1.058^{+0.068}_{-0.035}$
and $f_-^{D\to a_0}(0) = 0.764^{+0.044}_{-0.036}$. As a further application,
the branching fractions of the $D\to a_0(980)\ell\bar\nu_\ell$ semileptonic
decays are given. Taking the decay $a_0(980)\to \eta\pi$ into consideration, we
obtain ${\cal B}(D^0 \to a_0(980)^- (\to \eta \pi^-) e^+\nu_e)
=(1.330^{+0.216}_{-0.134})\times10^{-4}$, ${\cal B}(D^+\to a_0(980)^0(\to \eta
\pi^0)e^+\nu_e)=(1.675^{+0.272}_{-0.169})\times10^{-4}$, which are consistent
with the BESIII collaboration and PDG data within errors. Finally, we present
the angle observables of forward-backward asymmetries, $q^2$-differential flat
terms and lepton polarization asymmetry of the semileptonic decay $D\to
a_0(980)\ell\bar\nu_\ell$.
