We numerically investigate statistically steady quantum turbulence in the cyclic phase of a spin-2 spinor Bose-Einstein condensate (BEC) driven by large-scale energy injection via a divergence-free external velocity field. In the full non-Abelian cyclic system, the mass-current spectrum exhibits an anomalous k^{-7/3} scaling, whereas the spin-current spectrum approximately follows a k^{-5/3} power law. To isolate the role of non-Abelian vortex dynamics, we compare this system with an Abelian-restricted cyclic system sharing the same Hamiltonian parameters and driving conditions. Strikingly, the Abelian restriction collapses both the mass- and spin-current spectra into a k^{-5/3} scaling, identical to that of a scalar BEC, despite the three distinct turbulent states maintaining nearly identical vortex-line densities. A Helmholtz decomposition reveals that the mass- and spin-current spectra are predominantly incompressible and transverse, respectively, thereby confirming that the anomalous scaling cannot be attributed simply to compressible fluctuations or to the transverse nature of the energy injection. Local spectral exponents further quantify these distinct scaling regimes. The disappearance of the mass-spin spectral separation under the Abelian restriction, alongside the formation of a large-scale web of rung-connected non-Abelian vortices, provides compelling evidence that non-Abelian vortex dynamics fundamentally govern the organization of turbulent mass flow.