Max-pressure (MP) control has proven effective at stabilizing network queues and improving traffic throughput in large-scale urban road networks. However, conventional MP controllers based on connected vehicle (CV) data face two critical limitations: network stability diminishes when connected vehicle (CV) penetration rates are low, and significant privacy concerns arise when utilizing individual vehicle data. To address these challenges, this paper proposes a novel Private-MP controller that fuses data from both fixed-location detectors and CVs in an architecture of mobile edge computing. To fully safeguard CV privacy, including macro-route information and micro-trajectory information, Private-MP employs a privacy-preserving mechanism that combines homomorphic encryption with an adaptive randomized response strategy. Simulation studies on a network with five intersections showed that despite some increases in average vehicle delay due to privacy protection, Private-MP still ensures a more robust performance on average vehicle delay than CV-based MP in low penetration rate scenarios and outperforms traditional detector-based MP control while improving fairness among connected and non-connected vehicles.