Imaging with low-dose light is of importance in various fields, especially when minimizing radiation-induced damage onto samples is desirable. The raw image captured at the detector plane is then predominantly a Poisson random process with Gaussian noise added due to the quantum nature of photo-electric conversion. Under such noisy conditions, highly ill-posed problems such as phase retrieval from raw intensity measurements become prone to strong artifacts in the reconstructions; a situation that deep neural networks (DNNs) have already been shown to be useful at improving. Here, we demonstrate that random phase modulation on the optical field, also known as coherent modulation imaging (CMI), in conjunction with the phase extraction neural network (PhENN) and a Gerchberg-Saxton-Fienup (GSF) approximant, further improves resilience to noise of the phase-from-intensity imaging problem. We offer design guidelines for implementing the CMI hardware with the proposed computational reconstruction scheme and quantify reconstruction improvement as function of photon count.