We discuss the dependence of the eigenvalues and eigenfunctions for the discrete signed $p$-Laplacian under perturbation by a cut parameter. In particular, we prove a formula for the derivative of the eigenvalues and show that the eigenvalues of the discrete signed $p$-Laplacian on the original graph can be characterized via extremal points of the perturbed system. In this context, we elaborate on how graph surgery can be used in order to compute eigenvalues of the discrete (signed) $p$-Laplacian by looking at some examples. The derivation formula is reminiscent of the formula for linear eigenvalue problems given by the Hellmann-Feynman theorem and our results extend previous results for the linear case $p=2$ attained by [Berkolaiko, Anal. PDE 6 (2013), no. 5, 1213-€“1233].