Vector diffraction theory is used to investigate the focusing properties of cosh-Gaussian beams with the power-exponent-phase vortex. The effects of the decentered parameter, the power order, and the topological charge on the normalized intensity distribution are examined. Results show that intensity distribution in the focal region can be altered significantly by the topological charge, the power order, and the decentered parameter. The pattern of the optical intensity slowly enlarges with the increase of topological charge. The strongest intensity part of the annulus rotates by the changing topological charge. As the power order increases, the intensity distribution is more concentrated. As the decentered parameter increases, there occur multiple relatively strong intensity peaks, and the entire focus pattern extends outward. Some optical gradient force distributions are investigated to illuminate the applications of these alterable foci patterns.