Amplitude Analysis of ${D}_{s}^{+}\to {\pi }^{+}{\pi }^{0}\eta $ and First Observation of the $W$-Annihilation Dominant Decays ${D}_{s}^{+}\to {a}_{0}\left(980{\right)}^{+}{\pi }^{0}$ and ${D}_{s}^{+}\to {a}_{0}\left(980{\right)}^{0}{\pi }^{+}$

We present the first amplitude analysis of the decay ${D}_{s}^{+}\to {\pi }^{+}{\pi }^{0}\eta $. We use an ${e}^{+}{e}^{-}$ collision data sample corresponding to an integrated luminosity of $3.19\text{}\text{}{\mathrm{fb}}^{-1}$ collected with the BESIII detector at a center-of-mass energy of 4.178 GeV. We observe for the first time the $W$-annihilation dominant decays ${D}_{s}^{+}\to {a}_{0}\left(980{\right)}^{+}{\pi }^{0}$ and ${D}_{s}^{+}\to {a}_{0}\left(980{\right)}^{0}{\pi }^{+}$. We measure the absolute branching fraction $B\left({D}_{s}^{+}\to {a}_{0}\left(980{\right)}^{+\left(0\right)}{\pi }^{{0}^{\left(}+\right)},{a}_{0}\left(980{\right)}^{+\left(0\right)}\to {\pi }^{+\left(0\right)}\eta \right)=\left(1.46±0.1{5}_{\mathrm{stat}}±0.2{3}_{\mathrm{sys}}\right)%$, which is larger than the branching fractions of other measured pure $W$-annihilation decays by at least one order of magnitude. In addition, we measure the branching fraction of ${D}_{s}^{+}\to {\pi }^{+}{\pi }^{0}\eta $ with significantly improved precision.