Let F be a Siegel cusp form of integral weight k on the Siegel modular group Sp₂(ℤ) of genus 2. The coefficients of the spinor zeta function $Z_F(s)$ are denoted by cₙ. Let $D_ρ(x;Z_F)$ be the Riesz mean of cₙ. Kohnen and Wang obtained the truncated Voronoï-type formula for $D_ρ(x;Z_F)$ under the Ramanujan-Petersson conjecture. In this paper, we study the higher power moments of $D_ρ(x;Z_F)$, and then derive an asymptotic formula for the hth (h=3,4,5) power moments of $D₁(x;Z_F)$ by using Ivić's large value arguments and other techniques.