Safe navigation in unknown and cluttered environments remains a challenging problem in robotics. Model Predictive Contour Control (MPCC) has shown promise for performant obstacle avoidance by enabling precise and agile trajectory tracking, however, existing methods lack formal safety assurances. To address this issue, we propose a general Control Lyapunov Function (CLF) and Control Barrier Function (CBF) enabled MPCC framework that enforces safety constraints derived from a free-space corridor around the planned trajectory. To enhance feasibility, we dynamically adapt the CBF parameters at runtime using a Soft Actor-Critic (SAC) policy. The approach is validated with extensive simulations and an experiment on mobile robot navigation in unknown cluttered environments.