The singly Cabibbo-suppressed decay $D_{s}^{+} o K^+pi^{+}pi^{-}pi^{0}$
is observed for the first time, using a data set corresponding to an integrated
luminosity of 6.32~$
m fb^{-1}$ recorded by the BESIII detector at the
centre-of-mass energies between 4.178 and 4.226~GeV. The first amplitude
analysis of $D_{s}^{+} o K^+pi^{+}pi^{-}pi^{0}$ reveals the sub-structures
in this decay and determines the fractions and relative phases of different
intermediate processes. The dominant intermediate process is $D_s^+ o
K^{*0}
ho^+$, with a fit fraction of
$(40.9pm2.8_{
m{stat.}}pm1.5_{
m{syst.}})\%$. With the detection efficiency
based on our amplitude analysis, the absolute branching fraction for $D_{s}^{+}
o K^+pi^{+}pi^{-}pi^{0}$ is measured to be
$(9.75pm0.54_{
m{stat.}}pm0.17_{
m{syst.}}) imes 10^{-3}$.
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