This work deals with the study of a class of nonlocal Navier boundary value problems involving the degenerate $p(x)$-biharmonic operator with a potential term $q(x)$-Hardy \[ \begin{cases} \Delta(mega (|\Delta u|^{p(x)})|\Delta u|^{p(x)-2}\Delta u)-ambda\frac{|u|^{q(x)-2}u}{{ẹlta(x)}^{2q(x)}}=\mu ǎrtheta(x)|u|^{q(x)-2}u\bigg(ıt_{mega}\frac{ǎrtheta(x)}{q(x)}|u|^{q(x)}dx\bigg)^{r} & ext{in }mega, u=\Delta u=0, & ext{on } tialmega. \end{cases} \] In this new setting, our objectif is to extend the results obtained in the paper [M. Laghzal, A. El Khalil, M. D. Morchid Alaoui, A. Touzani, <i>Eigencurves of the $p(\cdot)$-biharmonic operator with a Hardy-type term potential</i>, Moroccan J. Pure Appl. Anal. <b>6</b>(2) (2020), 198-209] for the nonhomogeneous case $p(x)\neq q(x),$ where $\vartheta$ is a weight function. The main results are established by using the variational method and min-max arguments based on Ljusternik-Schnirelmann theory on $C^1$ manifoleds [A. Szulkin, <i>Schnirelmann theory on $C^1$-manifolds</i>, Ann. Inst. Henri Poincaré C, Anal. Non Linéaire <b>5</b>(2) (1988), 119-139]. A direct characterization of the principal curve (first one) is provided.