Efficient traffic signal control is essential for reducing congestion and improving urban mobility. Recently, learning-based methods have shown promise; however, many remain limited by high complexity, low interpretability, or poor generalization across diverse intersection types. This paper introduces GenGPLight, a generalized framework for optimizing traffic signal control across intersections with arbitrary structures, sizes, and signal configurations. GenGPLight is a learning-based optimization approach that uses genetic programming to automatically synthesize urgency functions for adaptive signal timing, which adjusts timings based on queue lengths detected by sensors or cameras. It leverages a flexible phase-level feature representation derived from aggregated lane-level traffic metrics, enabling the evolution of interpretable urgency-based control strategies. The framework is evaluated on real-world traffic networks, including both structured and unstructured scenarios, and benchmarked against established baselines. Results show that GenGPLight achieves robust performance, improved traffic efficiency, and strong generalization, highlighting its suitability for deployment in complex urban environments.