  From the amplitude analysis of the $D^+_s \to \pi^+ \pi^0 \eta$ decay, the
BESIII Collaboration firstly observed the $D^+_s \to a_0(980)^+\pi^0$ and
$D^+_s \to a_0(980)^0\pi^+$ decay modes, which are expected to occur through
the pure $W$-annihilation processes. The measured branching fraction
$\mathcal{B}[D_{s}^{+}\to a_{0}(980)^{+(0)}\pi^{0(+)},a_{0}(980)^{+(0)}\to
\pi^{+(0)}\eta]$ is, however, found to be larger than those of known
$W$-annihilation decays by one order of magnitude. This apparent contradiction
can be reconciled if the two decays are induced by internal $W$-conversion or
external $W$-emission mechanisms instead of $W$-annihilation mechanism. In this
work, we propose that the $D^+_s$ decay proceeds via both the external and
internal $W$-emission instead of $W$-annihilation mechanisms. In such a
scenario, we perform a study of the $D^+_s \to \pi^+\pi^0\eta$ decay by taking
into account the contributions from the tree diagram $D^+_s \to \rho^+ \eta \to
\pi^+ \pi^0 \eta$ and the intermediate $\rho^+ \eta$ and
$K^*\bar{K}/K\bar{K}^*$ triangle diagrams. The intermediate $a_0(980)$ state
can be dynamically generated from the final state interactions of coupled $K
\bar{K}$ and $\pi \eta$ channels, and it is shown that the experimental data
can be described fairly well, which supports the interpretation of $a_0(980)$
as a molecular state.
