The effects of defocus and astigmatism aberration of the turbulent atmosphere on the intensity distribution of a focused vortex carrying Gaussian beam are investigated. The integral expression of the intensity distribution on the focal plane in slant path is derived from the extended Fresnel-Kirchhoff diffraction integral and the quadratic approximation of phase structure function. And the influences of the three factors, turbulent strength, the propagation distance and the topological charge of the initial beam, are investigated by numerical calculation. Our results show that in weak turbulent region, the effects of defocus and astigmatism aberration in turbulent atmosphere on the intensity distribution of a vortex carrying Gaussian beam are very small and can be ignored. However, in the middle turbulent region, the effects of defocus and astigmatism are pronounced. As the propagation distance and the turbulent strength increase, both aberrations cause the decrease of the intensity peak value, the spreading of the beam, and the stretching of the dark core. For the beams with single topological charge, under the influence of astigmatism, the peak value of the intensity decreases more seriously, the intensity of the secondary maximum stripe increases much faster, and the focal spot expands more than the effect of defocus aberration. Compared with single topological charge, the effects of defocus and astigmatism of vortex beam with double topological charge are more obvious. The focal spots are expanded, and the secondary maximum stripes are more obvious. However, under the effect of astigmatism, because of the reduction of the coherence and the deflection effect of the beam, the dark core on the focal plane tends to break into two dark regions separated by a bright region.