  We construct a non-trivial homomorphism from the Guay's affine Yangian
associated with $\widehat{\mathfrak{sl}}(n)$ to the universal enveloping
algebra of the $W$-algebra associated with a Lie algebra $\mathfrak{gl}(m+n)$
and its nilpotent element whose Jordan block of type $(2^{n},1^{m-n})$ for
$m>n$.
