Let \(\Delta^2_pu=\Delta(|\Delta u|^{p-2}\Delta u)\) be an fourth-order so-called \(p\)-biharmonic operator. In this article the existence of infinitely many weak solutions for the \(p\)-biharmonic equation with Hardy potential of Kirchhoff type \[ \begin{gathered} M\left(\int_\Omega|\Delta u|^p\,dx\right)\Delta^2_pu- \frac{a}{|x|^{2p}}|u|^{p-2}u=\lambda f(x,u)+\mu g(x,u),\quad x\in\Omega, \\ u=\Delta u=0,\quad x\in\partial\Omega \end{gathered}\tag{*} \] is considered. Here \(\Omega\) is a bounded domain in \(\mathbb R^N\) (\(N\geq3\)) containing the origin and with a smooth boundary \(\partial \Omega\), \(1<p<\frac{N}2\), \(\lambda>0\), \(\mu>0\), \(M:[0;+\infty)\to\mathbb R_+\) is a continuous function such that \(\exists\,m_0,m_1>0\) \(\forall\,t\geq0\) \(m_0\leq m(t) \leq m_1\) and \(f, g:\Omega\times\mathbb R\to\mathbb R\) are two continuous functions. If \(M(\cdot)=1\), then Kirchhoff type problem (*) reduces to the \(p\)-biharmonic equation with Hardy potential. Under some conditions on functions \(F(x,t)=\int_0^t f(x,\xi)\,d\xi\geq0\) and \(G(x,t)=\int_0^t g(x,\xi)\,d\xi\geq0\), \((x,t)\in\Omega\times\mathbb R\), for functions \(f,g\in C(\overline{\Omega} \times \mathbb R)\) and for \(\lambda\in \Lambda\), \(\mu\in[0,\mu_*)\) the problem (*) possesses an unbounded sequence of weak solutions in \(X=W^{2,p}(\Omega) \cap W^{1,p}_0(\Omega)\) endowed with the norm \(\|u\|^p=\int_\Omega| \Delta u|^p\,dx\) (Theorem 3.2). Similar result is established in Theorem 3.3. A sequence of solutions \(\{u_n\}\) obtained there weakly converges at zero.