A high-gain and beam-switchable multi-beam holographic metasurface antenna (HMSA) array coated with phase-correcting metasurface (PCMS) is proposed. It comprises four identical HMSAs arranged in a crossed-shaped structure. Each beam has a different direction to allow for switchable control according to engineering specifications. The HMSA pattern is generated through holographic interference between the desired beam and the reference wave. A top-hat monopole Yagi radiator is employed as the surface wave launcher to produce the reference wave. To enhance the gain, a PCMS is strategically suspended over the HMSA array at a distance of 6 mm. Each cell of the PCMS can provide arbitrary phase compensation in the reference plane to realize the in-phase plane wave. The phase lag arises from non-uniform phase distributions on the MS and different distances from each MS unit cell to the reference plane. It also acts as a superstrate to adjust the free-space wave coupling path. When the free-space wave and surface wave coupling have the same amplitude but out of phase, they subtract each other. A prototype is fabricated and measured, demonstrating that the use of PCMS can enhance the radiation performance of the HMSA array. This achieves a peak gain of 19.53 dBi, an SLL of less than -14.5 dB, and an aperture efficiency (AE) of more than 26.48%.