This study investigates a class of games with cooperative-competitive coupling characteristics. Incorporating dynamic environmental features, a multi-cluster game model with time-varying cost functions is constructed. Based on the time-varying communication network topology, a discrete-time distributed Nash equilibrium online learning algorithm is proposed, which optimizes communication resources through an event-triggered mechanism: 1) Preset event-triggered conditions are employed to regulate information interaction among agents; 2) The algorithm maintains Nash equilibrium solution accuracy while reducing communication frequency. Theoretical analysis demonstrates that by judiciously selecting time-varying step sizes ${\Big (}{{\alpha }_{t}}=\dfrac{1}{{{t}^{{2{a}_{1}}}}}, 0<a_1<\dfrac{1}{2}{\Big)} $ and event-triggered function thresholds $(C_0\alpha_t, C_0>0) $, the designed algorithm effectively mitigates transmission and estimation errors in time-varying environments, with its exponential convergence properties rigorously proven. Simulation experiments in a typical energy internet scenario show that compared to conventional algorithms, the proposed method effectively reduces communication frequency while ensuring Nash equilibrium convergence accuracy, validating its engineering efficacy.