The invention discloses a moving type single pile gripper construction mechanism with an automatic deviation rectifying function. The construction mechanism comprises a pile gripper base, wherein a pile gripper is arranged on the pile gripper base; gripping rings are arranged on the pile gripper; a centering block is arranged on each of the gripping rings; a wheel box is arranged on each of the centering blocks in a welding manner; a V-shaped mechanism is mounted on a pile gripper support; an inserting arm assembly is arranged under the pile gripper base; the inserting arm assembly is embedded in a barge; and a crane is arranged on the barge. The invention further discloses a pile gripper construction method. The construction method mainly comprises the following steps: pre-mounting a pile gripper assembly, moving the barge to a specific location, driving auxiliary gauge piles, removing the barge, starting the centering blocks, hoisting a steel pipe pile, performing cross type gripping, performing GPS positioning, driving the steel pipe pile and monitoring the perpendicularity thereof, performing pile sinking by adopting a hydraulic hammer, and moving the pile gripper assembly. The moving type single pile gripper construction mechanism and method with the automatic deviation rectifying function have the advantages as follows: accurate piling positioning is achieved, automatic deviation rectifying is realized, the process is convenient, the transportation is convenient, the efficiency is high, and the cost is lower.