Sparse arrays are a topic of high interest within the ultrasound (US) imaging community, because of their promising ability to reduce costs, complexity, energy consumption, and data transfer requirements of US systems, thus addressing the main challenges of 3-D and portable 2-D systems. Undersampling a transducer array usually results in a significant increase in imaging artifacts, caused primarily by higher grating lobe (GL) levels. Thus, state-of-the-art sparse arrays design strategies focus on avoiding GLs, while compromising on the resulting image resolution and uniformity. In this work, we investigated the applicability of convolutional neural network (CNN)-based image reconstruction, having recently proven its potential in reducing GL artifacts, for reconstructing images from single unfocused acquisitions using uniformly undersampled linear array configurations on receive. The proposed reconstruction method consists of first computing a low-quality estimate from the undersampled single-shot acquisitions using a delay-and-sum (DAS) algorithm, followed by applying a real-time-capable CNN, trained specifically to reduce diffraction artifacts. Experiments were conducted within a simulation environment, in the context of plane wave imaging on a numerical test phantom dedicated to US image quality assessment. The proposed approach achieved an image comparable or better to that obtained from conventional DAS beamforming using the full array with uniformly undersampled arrays up to a factor of three, demonstrating a promising potential for sparse array imaging in general.