This study investigates the propagation of flexural-gravity waves in an incompressible fluid of finite depth over an elastic bottom, influenced by a constant magnetic field. The fluid surface is covered by a thin sheet of floating ice, modeled as a thin elastic plate of uniform thickness, using the Euler-Bernoulli beam equation. An in-plane compressive force acting on the ice is also considered. The ocean bottom is represented as an elastic half-space using linear elasticity theory. The flexural-gravity wave model is formulated using linear water wave theory and small-amplitude structural response in finite water depth, while linear elasticity theory is employed to describe the elastic bed. The dispersion relation, phase velocity related to the wave motion are derived. Graphical analysis illustrates the influence of compressive force and ice flexural rigidity on wave characteristics. Additionally, the stress components acting on the elastic bottom and the ratio of ground displacement to the deflection of the ice-covered surface are evaluated.