= Solution
If $p\nmid n$, the preceding part says $U=S_F^{(n-1,1)}$ is simple and has no nonzero <homomorphism> into $W$. Its only nonzero simple <quotient> is itself, so $W$ cannot contain a <submodule> isomorphic to such a <quotient>.
If $p\mid n$, use the given nonzero <homomorphism> $\phi:U\to W$. The <submodule> chain shows that $T$ is the unique maximal proper <submodule> of $U$ and $U/T$ is simple. Moreover $\operatorname{Hom}_{S_n}(T,W)=0$: for odd <characteristic> this follows from part (b), and in <characteristic two> the divisibility condition makes $n$ even, where the corrected fixed-space calculation also gives zero. Thus $\phi$ kills $T$ and descends to a nonzero map $U/T\to W$. Simplicity makes this map <injective>.
Its <image> is the required simple <submodule>, proving
$$
\boxed{W\text{ contains a simple quotient of }U\iff p\mid n.}
$$
In the divisible case the <quotient> is explicitly $U/T$, of <dimension> $n-2$. The argument uses the supplied existence of $\phi$ and only the fixed-space vanishing that remains valid in that case.
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