In this paper we study polynomial Hamiltonian systems $$dF=0$$ in the plane and their deformations $$dF+\epsilon \omega =0$$, where $$\omega $$ is a polynomial 1-form. We consider the first nonzero Melnikov function, $$M_{\mu }$$, of the displacement function $$\Delta (t,\epsilon )=\sum _{j=\mu }^{\infty }\epsilon ^{j}M_{j}(t)$$, along a cycle $$\gamma (t)$$ in $$F^{-1}(t)$$. It is known, that in the generic case $$M_{\mu }$$ is an abelian integral (Francoise in Ergod Theory Dyn Syst 16(1):87–96, 1996; Ilyashenko in Mat Sb (N.S.) 78(120):360–373, 1969), and an iterated integral of length at most $$\mu $$ in general (Gavrilov in Ann Fac Sci Toulouse Math (6) 14(4):663–682, 2005). Here we study linear deformations of a family of non-generic Hamiltonians systems $$dF=0$$, where $$F=\prod _{j=1}^rf_j\in {{\mathbb {R}}}[x,y]$$, with $$f_j=f_{1j}^{n_j}+g_j$$, $$n_j\in {{\mathbb {N}}}$$, for $$f_{1j}$$$$(j=1,\ldots ,r)$$ pairwise linearly independent polynomials of degree one, and $$g_j$$ a polynomial of degree smaller than $$n_j$$ (Pontigo-Herrera in J Dyn Control Syst 23(3):597–622, 2017). We also assume some geometric properties on F; namely, $$\overline{F^{-1}(0)}$$ defines a good divide with r branches in $${{\mathbb {R}}}{{\mathbb {P}}}^2$$ (where only the zero critical level can have more than one critical point) and F has good multiplicity at infinity (A’Campo in Math Ann 213:1–32, 1975; Pontigo-Herrera in J Dyn Control Syst 23(3):597–622, 2017). We denote this family by $${\mathcal {F}}_r({{\mathbb {R}}})$$. We prove that for polynomials in $${\mathcal {F}}_r({{\mathbb {R}}})$$, the first nonzero Melnikov function of their deformations are iterated integrals of length at most two.