Complex pipeline network should be inspected regularly for safety. In general, these tasks are often completed by large pipe-climbing robots or customized equipment. Most of them are not effective, and cannot work on pipes with uncertain barriers. Moreover, some pipes are mounted in constrained scenarios, so bulky robots are not applicable. This paper presents a tethered soft-rigid pipe-climbing robot to fill the gap. Two indispensable actions, i.e. embracing pipe and moving along it, are realized by a soft component and a 3D printed wheel mechanism. The latter includes two forces: thrust force from the compressed air and tractive force from wheels, to drive the robot comprehensively, and a lightweight body (only 160g) benefits agile motion. In operation, pressure exerted on the soft component enables the robot to embrace pipes of different diameters, with controllable adhesion force, and locomotion force is also regulated. Inspired from vehicle, an elastic damper is attached between the wheel structure and the robot body, which can effectively alleviate vibration when crossing barriers. In addition, theoretical models are constructed to analyse and control thrust force, and the locomotion performance is analysed by dynamics model. Experiments demonstrate that this robot can perform rapid climbing at a speed of 1.09m/s in load-free scenarios, and it can move at a maximum speed of 0.828m/s with 500g load. Reconstruction of a flexible pipe using the robot is also demonstrated.