Asymptotically safe gravity is an effective approach to quantum gravity. It is important to differentiate modified gravity, which is inspired by asymptotically safe gravity. In this study, we examine particle dynamics near the improved version of a Schwarzschild black hole. We assume that in the context of an asymptotically safe gravity scenario, the ambient matter surrounding the black hole is of isothermal nature, and we investigate the spherical accretion of matter by deriving solutions at critical points. The analysis of various values of the state parameter for isothermal test fluids, viz., k = 1, 1/2, 1/3, 1/4 show the possibility of accretion onto an asymptotically safe black hole. We formulate the accretion problem as Hamiltonian dynamical system and explain its phase flow in detail, which reveals interesting results in the asymptotically safe gravity theory.