Zonal flow and turbulence damping are two common physical phenomena in the evolution of plasma flows, and nonlinear energy transfer in turbulence is the fundamental mechanism of flow driving. Based on HL-2A tokamak edge plasma turbulence and zonal flow experiments, this study employs the two-point method to measure floating potential signal data and utilizes bispectral analysis for data processing. From the perspective of nonlinear energy function analysis, it revealed the physical picture and mechanisms of coexisting nonlinear dissipation and nonlinear energy transfer in zonal flows and turbulence in tokamak edge plasmas. The findings indicate that turbulence can evolve and develop within fluid systems featuring background dissipation. Notably, a significant damping of zonal flows is observed. The research uncovers an energy cascade process from geodesic acoustic mode (GAM) zonal flows to high-frequency turbulence, where GAM-induced turbulent fluctuations cause internal wave energy to transfer from low- to high-frequency regions. In another discharge type, the coexisting phenomena of turbulence damping and energy cascade are observed, as shown in figure A. Energy cascade transforms large vortices into small ones, ultimately dissipating linearly. In both cases, only a portion (approximately 30%-40%) of nonlinear energy participates in energy transfer, while the majority dissipates due to nonlinear damping effects of zonal flows and turbulence. Note that the estimated ratio values mentioned above may vary depending on the specific discharge parameters. This demonstrates that turbulence can naturally decay through viscous and collisional damping. Collisional and other damping effects can trigger nonlinear energy dissipation. Additionally, the inverse energy cascade phenomena are observed in HL-2A edge plasma turbulence, accompanied by a pronounced turbulence damping. When the energy fraction involved in inverse energy cascade is limited, the inverse energy cascade in turbulence becomes insufficient to drive large-scale structures like zonal flows. The study reveals that the energy dissipation of zonal flow and turbulence includes two forms. The energy cascade in turbulence can convert a portion of large eddies into small eddies, which then dissipate linearly. Meanwhile, due to collisional damping and other damping effects, most low-frequency energy directly dissipates nonlinearly.