<p>This paper focuses on a (2+1)-dimensional fourth-order nonlinear wave equation with five categories of nonlinear terms, which can be reduced to spatially symmetric nonlinear models. Rational and lump solutions are derived by taking suitable limits of the soliton solutions obtained via the Hirota bilinear method. Furthermore, a kind of specific <i>N</i>-soliton solutions satisfying certain constraints is also obtained. Additionally, a specialized spatially symmetric model is presented to explore the corresponding lump waves. The derived lump solutions feature a critical point line, where their two spatial coordinates move at a constant velocity. Moreover, a reduced case is computed, revealing that nonlinearity and dispersion jointly control lump waves. This work enriches the solution structure of high-dimensional nonlinear equations and provides insights for describing complex dispersive phenomena.</p>