In a previous Memoirs (Vol. 92, No. 448), Levenberg and Yamaguchi analysed the second variation of the Robin function $-\lambda(t)$ associated to a smooth variation of domains in $\mathbb{C}^n$ for $n\geq 2$. In the current work, the authors study a generalisation of this second variation formula to complex manifolds $M$ equipped with a Hermitian metric $ds^2$ and a smooth, nonnegative function $c$.