In this paper, the dynamics of the circular Airy beam (CAB) in the spatial fractional nonlinear Schrödinger equation (FNLSE) optical system are investigated. The propagation characteristics of CABs modulated by the quadratic phase modulation (QPM) in a Kerr (cubic) nonlinear medium under power function diffractive modulation modes and parabolic potentials are numerically simulated by using a step-by-step Fourier method. Specifically, the threshold for CABs to form solitons in the Kerr medium is controlled by the Lévy index and the QPM coefficient. Secondly, the parabolic potential has the ability to stabilize the FNLSE optical system, making it easier for the formation of CAB solitons. The addition of QPM allows the refocusing of the split beam caused by the Lévy index, and it can change the position and intensity of solitons. Finally, we also study the transmission evolution of QPM-modulated CABs in the Kerr medium under the power function diffraction modulation mode. We can obtain different types of solitons by varying the power function modulation coefficients. A dark soliton with high stability is formed, and we can control its size. Results show that it is possible to optimize the parameter settings (parabolic potential coefficients, power function modulation coefficients, QPM coefficients, Lévy indices, and nonlinear Kerr intensity coefficients) to obtain different types of solitons as well as to modulate the soliton transport. It provides more degrees of freedom for the study of CAB soliton propagation in the Kerr media, which is of great significance and application in fields of nonlinear optical transport, particle manipulation, and optical metrology.