In this paper, the circular Bessel Gaussian beams (CBGBs) carrying power-cotangent-phase vortices are firstly introduced, whose propagation dynamics are explored theoretically and experimentally. The number of spiral lobes, rotation direction, rotation angle, and shape of the new type of beam can be flexibly modulated by controlling multiple parameters of power-cotangent-phase vortices. Accordingly, the effect of multiple beam parameters on abruptly autofocusing ability is quantified and compared by using the K-value curve that is described by ratio Im/I0, where Im and I0 correspond to the maximum intensities at different propagation distance and the initial plane, respectively. The physical mechanism of intensity distribution variation depended on the propagation distance and power-cotangent-phase parameters are also demonstrated convincingly by employing the Poynting vector. In addition, the advantages and applications of the proposed beam as a tool for the Rayleigh particle manipulation are analyzed theoretically. It is expected that the introduced beam can be useful for extending applications of optical vortices, particularly for multiple particle manipulation.