Poisson image deconvolution and denoising have received much attention in recent years. A key challenge of this problem is that the most effective prior might never exist. The purpose of our study is to investigate the characteristics of generalized l_p  /  l_q norm on the derivatives of nature images and to present an efficient method to promote hyper-Laplacian prior for the deconvolution of blurry and Poisson noisy images. We first analyze the mathematical characteristics of generalized l_p  /  l_q norm based on the Berkeley segmentation data set and implement deconvolution using generalized l_p  /  l_q norm for derivatives in high-frequency domain so as to enforce the preservation of strong edges. Then hyper-Laplacian prior is promoted for actual Poisson image deconvolution, which uses the estimated gradient information to penalize the small derivatives (often noise) and preserve large derivatives associated to image borders. Our preliminary experiments show that the proposed method can achieve higher image quality than state-of-the-art methods in terms of peak signal-to-noise ratio and noise sensitivity.