Based on the extended Huygens-Fresnel principle and the quadratic approximation of phase structure function, the theoretical modal of optical intensity distribution on the focal plane for focused hollow vortex beams with a Gaussian background propagating in turbulent atmosphere was investigated and obtained.The effects of atmospheric index structure constant C2n, topological charge, focal length and beam wavelength on optical intensity distribution on the focal plane were analyzed. The results indicate that with the increase of focal length, the intensity distribution on the focal plane varies from central-dip form to Gaussian distribution. The effects of weak turbulence on intensity distribution are negligible. The maintenance of singularity of vortex beams with larger topological charge in turbulent atmosphere is better than that for vortex beams with smaller topological charge. The central-dip form of the intensity distribution becomes shallow and the intensity distribution becomes smooth with the turbulence outer scale increasing.