  We perform for the first time an amplitude analysis of the decay $D^{+}\to
K_{S}^{0}\pi^+\eta$ and report the observation of the decay $D^{+}\to
K_{S}^{0}a_{0}(980)^{+}$ using 2.93 fb$^{-1}$ of $e^+e^-$ collision data taken
at a center-of-mass energy of 3.773 GeV with the BESIII detector. As the only
W-annihilation free decay among $D$ to $a_{0}(980)$-pseudoscalar, $D^{+}\to
K_{S}^{0}a_{0}(980)^{+}$ is the ideal decay to extract the contributions of the
external and internal $W$-emission amplitudes involving $a_{0}(980)$ and study
the final-state interactions. The absolute branching fraction of $D^{+}\to
K_{S}^{0}\pi^+\eta$ is measured to be $(1.27\pm0.04_{\rm stat.}\pm0.03_{\rm
syst.})\%$. The product branching fractions of $D^{+}\to
K_{S}^{0}a_{0}(980)^{+}$ with $a_{0}(980)^{+}\to \pi^+\eta$ and $D^{+}\to \pi^+
K_0^*(1430)^0$ with $K_0^*(1430)^0\to K_{S}^{0}\eta$ are measured to be
$(1.33\pm0.05_{\rm stat.}\pm0.04_{\rm syst.})\%$ and $(0.14\pm0.03_{\rm
stat.}\pm0.01_{\rm syst.})\%$, respectively.
