For a bounded domain $Ω\subset\mathbb{R}^n$ let $H_Ω:Ω\timesΩ\to\mathbb{R}$ be the regular part of the Dirichlet Green function for the Laplace operator. Given a fixed arbitrary ${\mathcal C}^2$ function $f:{\mathcal D}\to\mathbb{R}$, defined on an open subset ${\mathcal D}\subset\mathbb{R}^{nN}$, and fixed coefficients $λ_1,\dots,λ_N\in\mathbb{R}\setminus\{0\}$ we consider the function $f_Ω:{\mathcal D}\capΩ^N\to\mathbb{R}$ defined as \[ f_Ω(x_1,\dots,x_N) = f(x_1,\dots,x_N) - \sum_{j,k=1}^N λ_jλ_k H_Ω(x_j,x_k). \] We prove that $f_Ω$ is a Morse function for most domains $Ω$ of class ${\mathcal C}^{m+2,α}$, any $m\ge0$, $0

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