In this paper, we propose a real-time control method for unmanned aerial vehicles to execute missions involving time-sensitive targets (TSTs) in dynamic environments with obstacles. The nonlinear dynamics inherent to unmanned aerial vehicles (UAVs) often limit the effectiveness of traditional control methods that rely on simplified models, leading to infeasibility in practical applications. In TST missions, stringent temporal constraints further restrict maneuverability, while the presence of obstacles narrows the feasible control space and increases the difficulty of achieving timely and safe task completion. To address these challenges, We integrate Signal Temporal Logic and Linear Temporal Logic to formalize complex task requirements, and utilize control barrier functions to enforce real-time safety constraints. These components are unified with a quadratic programming framework that generates continuous control inputs for nonlinear systems. Simulation results validate the effectiveness of the proposed method in satisfying time constraints, ensuring obstacle avoidance, and enabling adaptive and feasible control in dynamic scenarios.