Continuum robots are well-suited for constrained environments due to their superior flexibility and structural compliance. However, relying solely on passive compliance may lead to damage to both the robot and the surrounding environment. This work proposes a finite-time Cartesian impedance control scheme for tendon-driven continuum manipulators (TDCMs), where a second-order low-pass filter is used to adjust the reference trajectory according to the external robot tip force. The controller is designed using the command filtered backstepping method, and the finite-time stability is established by the designed Lyapunov function. In TDCM systems, the tendons operate antagonistically, and actuators often fail to quickly reach the desired tendon tension, leading to partial failures. To address this, we propose an actuator fault compensation algorithm to enhance system performance and reliability. We conducted trajectory tracking experiments on a multi-segment TDCM prototype, the results demonstrate that the designed Cartesian impedance controller achieves effective compliance control effect and high position control accuracy.