In this study, the vortex-shaped photoelectron momentum distributions of a hydrogen atom ionized by two time-delayed circularly polarized laser fields are numerically simulated based on the strong field approximation (SFA) theory. Under the action of two time-delayed laser pulses, the electron absorbs photons to overcome the ionization threshold and undergoes photoionization via two different transition channels to reach the continuum states for electron wave-packet interference. The simulation results show that the orientation of the photoelectron momentum vortices is observed to be related with the polarization directions of the two pulses, whereas the number of vortex arms is related with the carrier frequencies. Furthermore, the dynamic Stark effect is a ubiquitous strong field process, which would result in the distortion of the vortex-shaped momentum distributions when considered. On this basis, the clockwise momentum vortices and their distortion are specifically investigated, revealing that the distortion is attributed to the nonlinear properties of the phase associated with the dynamic Stark effect.