The paper focuses on the existence and multiplicity of weak solutions to nonlinear Kirchhoff-type equations involving $\psi$-Hilfer derivatives with $p(\cdot)$-Laplacian operators and Dirichlet boundary conditions. Through the application of a critical point approach, along with genus theory and variational techniques, we establish the existence and multiplicity results within appropriate fractional $\psi$-Hilfer derivative spaces. Our novel main results contribute to the advancement of the literature on differential equations involving fractional $\psi$-Hilfer generalized curvature phenomena.