We present a method for single-shot blind deconvolution in coherent diffraction imaging. Coherent diffraction imaging is a technique for non-interferometric quantitative phase imaging without reference light. In our method based on coherent diffraction imaging, a complex amplitude object is illuminated with coherent light, and light from the object is captured through unknown aberrating media and a coded aperture located on the pupil plane to reduce estimated variables on the aberrated pupil function. Both the amplitude and the phase of the object are recovered from the single captured intensity image by a phase retrieval algorithm in which the coded aperture is utilized as a support to estimate the sparse aberrated pupil function. We numerically and experimentally demonstrate the proposed method with complex amplitude objects under severe aberrating conditions. In the experiment, we quantitatively evaluate its performance with ptychography, which is a method for multi-shot coherent diffraction imaging. Our method enables quantitative phase imaging through turbulence by using simple and reference-free optical hardware without any invasive process.