  In this paper, we make a detailed discussion on the $\eta$-meson
leading-twist light-cone distribution amplitude (LCDA) $\phi_{2;\eta}(u,\mu)$
by using the QCD sum rules approach under the background field theory. Taking
both the non-perturbative condensates up to dimension-six and the
next-to-leading order (NLO) QCD corrections to the perturbative part, its first
three moments $\langle\xi^n_{2;\eta}\rangle|_{\mu_0} $ with $n = (2, 4, 6)$ can
be determined, where the initial scale $\mu_0$ is set as the usual choice of
$1$ GeV. Numerically, we obtain $\langle\xi_{2;\eta}^2\rangle|_{\mu_0}
=0.204_{-0.012}^{+0.008}$, $\langle\xi_{2;\eta}^4 \rangle|_{\mu_0} =0.092_{ -
0.007}^{ + 0.006}$, and $\langle\xi_{2;\eta}^6 \rangle|_{\mu_0}
=0.054_{-0.005}^{+0.004}$. Next, we calculate the $D_s\to\eta$ transition form
factor (TFF) $f_{+}(q^2)$ within the QCD light-cone sum rules approach up to
NLO level. Its value at the large recoil region is $f_{+}(0) =
0.484_{-0.036}^{+0.039}$. After extrapolating the TFF to the allowable physical
region, we then obtain the total decay widthes and the branching fractions of
the semi-leptonic decay $D_s^+\to\eta\ell^+ \nu_\ell$, i.e. $\Gamma(D_s^+
\to\eta e^+\nu_e)=(31.197_{-4.323}^{+5.456})\times 10^{-15}~{\rm GeV}$, ${\cal
B}(D_s^+ \to\eta e^+\nu_e)=2.389_{-0.331}^{+0.418}$ for $D_s^+ \to\eta
e^+\nu_e$ channel, and $\Gamma(D_s^+
\to\eta\mu^+\nu_\mu)=(30.849_{-4.273}^{+5.397})\times 10^{-15}~{\rm GeV}$,
${\cal B}(D_s^+ \to\eta\mu^+\nu_\mu)=2.362_{-0.327}^{+0.413}$ for $D_s^+
\to\eta\mu^+\nu_\mu$ channel respectively. Those values show good agreement
with the recent BES-III measurements.
