Human--robot co-carrying tasks demonstrate their potential in both industrial and everyday applications by leveraging the strengths of both parties. Effective control of robots in these tasks requires minimizing position and velocity errors to complete the shared tasks while also managing the energy level within the closed-loop systems to prevent potential dangers such as instability and unintended force exertion. However, this collaboration scenario poses numerous challenges due to varied human intentions in adapting to workspace characteristics, leading to human--robot conflicts and safety incidents. In this paper, we develop a robot controller that enables the robot partner to re-plan its path leveraging conflict information, follow co-carrying motions accurately, ensure passivity, and regular the energy of the closed-loop system. A cooperation control framework for human--robot co-carrying tasks is constructed by utilizing admittance control and time-varying Passive Velocity Field Control with a fractional exponent energy compensation control term. By measuring the interaction force, the desired trajectory of co-carrying tasks for the robot partner is first generated using admittance control. Thereafter, the new Passive Velocity Field Control with the energy compensation feature is designed to track the desired time-varying trajectory and guarantee passivity. Furthermore, the proposed approach ensures that the system's kinetic energy converges to the desired level within a finite time interval, which is critical for time-critical applications. Numerical simulation demonstrates the efficiency of the proposed cooperation control method through four collaborative transportation scenarios.