  In this paper, Lie super-bialgebra structures on a class of generalized super
$W$-algebra $\mathfrak{L}$ are investigated. By proving the first cohomology
group of $\mathfrak{L}$ with coefficients in its adjoint tensor module is
trivial, namely, $H^1(\mathfrak{L},\mathfrak{L}\otimes {\mathfrak{L}})=0$, we
obtain that all Lie super-bialgebra structures on $\mathfrak{L}$ are triangular
coboundary.
