Imaging a target scene with specular surfaces is a daunting challenge for both direct imaging and indirect computational imaging techniques. The intense specular reflection component during the measurement severely degrades the quality of the reconstructed image, resulting in a substantial loss of scene information. To address this issue, we propose a computational ghost imaging (CGI) method with adaptive intensity illumination. Capitalizing on the encoded imaging feature of CGI, this method enables effective imaging of target scenes with specular surfaces through two series of measurements, eliminating the necessity for additional optical components. Based on the position and intensity information of pixels in the specular regions from the first series of measurements, our method modulates the illumination patterns to weaken the intensity of the specular region in the second series of measurements. Simulation and experimental results demonstrate that the utilization of these modulated illumination patterns for target scene measurement effectively mitigates interference from the specular surface during imaging. Consequently, the reconstructed image is capable of presenting more detailed information about the target scene other than the specular regions. Our work introduces a novel approach for imaging target scenes with specular surfaces and broadens the scope of applications for CGI in reality.