In this Letter, we introduce a new class of angular dependent autofocusing ring Pearcey beams by imposing a cross-phase structure. Due to this structure, the beam exhibits a non-uniform abrupt autofocusing behavior. Unlike the properties of the ring Pearcey beam without a cross phase [Opt. Lett.43, 3626 (2018)OPLEDP0146-959210.1364/OL.43.003626], we can flexibly adjust the focal length of the beam and its focusing ability, as well as the direction of the ring Pearcey beams, with the help of only the cross-phase structure. Furthermore, the Poynting vectors are employed to demonstrate convincingly the beam-focusing mechanism. Such beams with these fascinating characteristics are anticipated to find potential applications in optical tweezing, three-dimensional printing, material processing, and so on.