  We consider a multivalued function of the form
$H\_{\varepsilon}=P\_{\varepsilon}^{\alpha\_0}\prod^{k}\_{i=1}P\_i^{\alpha\_i},
P\_i\in\mathbb{R}[x,y], \alpha\_i\in\mathbb{R}^{\ast}\_+$, which is a Darboux
first integral of polynomial one-form
$\omega=M\_{\varepsilon}\frac{dH\_{\varepsilon}}{H\_{\varepsilon}}=0,
M\_{\varepsilon}=P\_{\varepsilon}\prod^{k}\_{i=1}P\_i$. We assume, for
$\varepsilon=0$, that the polycyle $\{H\_0=H=0\}$ has only cuspidal singularity
which we assume at the origin and other singularities are saddles.
  We consider families of Darboux first integrals unfolding $H\_{\varepsilon}$
(and its cuspidal point) and pseudo-Abelian integrals associated to these
unfolding. Under some conditions we show the existence of uniform local bound
for the number of zeros of these pseudo-Abelian integrals.
