In this paper, cooperative energy recycling (CER) is investigated in wireless-powered mobile edge computing systems. Unlike conventional architectures that rely solely on a dedicated power source, wireless sensors are additionally enabled to recycle energy from peer transmissions. To evaluate system performance, a joint computation optimization problem is formulated that integrates local computing and computation offloading, under an alpha-fairness objective that balances total computable data and user fairness while satisfying energy, latency, and task size constraints. Due to the inherent non-convexity introduced by coupled resource variables and fairness regularization, a variable-substitution technique is employed to transform the problem into a convex structure, which is then efficiently solved using Lagrangian duality and alternating optimization. To characterize the fairness-efficiency tradeoff, closed-form solutions are derived for three representative regimes: zero fairness, common fairness, and max-min fairness, each offering distinct system-level insights. Numerical results validate the effectiveness of the proposed CER-enabled framework, demonstrating significant gains in throughput and adaptability over benchmark schemes. The tunable alpha fairness mechanism provides flexible control over performance-fairness trade-offs across diverse scenarios.