We show that every perturbation $A(\lambda, \epsilon)$ of an $n \times n$ matrix polynomial $A(\lambda)$ such that $\det A(\lambda) = \lambda^m$ with $m \le n$ can be reduced by equivalence transforms to a perturbed matrix polynomial whose leading matrix has maximal Smith form. This yields a reduced form for square perturbed matrix polynomials from which one can easily recover all the eigenvalue leading terms of the form $\mu \epsilon^\beta$ with $\beta^{-1}\in\mathbb{N}^*$.