This paper aims to develop an efficient and safe navigation framework under relaxable constraints. However, constraint-based navigation planning is challenging since the structure of the navigation environment and the existence of relaxable constraints are difficult to predict in advance. Furthermore, constraint relaxation priority varies depending on operation and task conditions. To address this, we introduce a novel reactive constraint-relaxation-and-planning method that autonomously identifies a constrained region to violate and redetermines a navigation plan that balances safety and efficiency in real time. We validate our approach through both quantitative and qualitative analysis using CARLA simulations, achieving a navigation success rate of 95% and reducing the navigation distance in the constraint region by 60% compared to scenarios with no constraint relaxation.