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Integrated photonic neural networks (IPNNs) are emerging as promising successors to conventional electronic AI accelerators as they offer substantial improvements in computing speed and energy efficiency. In particular, coherent IPNNs use arrays of Mach-Zehnder interferometers (MZIs) for unitary transformations to perform energy-efficient matrix-vector multiplication. However, the underlying MZI devices in IPNNs are susceptible to uncertainties stemming from optical lithographic variations and thermal crosstalk and can experience imprecisions due to non-uniform MZI insertion loss and quantization errors due to low-precision encoding in the tuned phase angles. In this paper, we, for the first time, systematically characterize the impact of such uncertainties and imprecisions (together referred to as imperfections) in IPNNs using a bottom-up approach. We show that their impact on IPNN accuracy can vary widely based on the tuned parameters (e.g., phase angles) of the affected components, their physical location, and the nature and distribution of the imperfections. To improve reliability measures, we identify critical IPNN building blocks that, under imperfections, can lead to catastrophic degradation in the classification accuracy. We show that under multiple simultaneous imperfections, the IPNN inferencing accuracy can degrade by up to 46%, even when the imperfection parameters are restricted within a small range. Our results also indicate that the inferencing accuracy is sensitive to imperfections affecting the MZIs in the linear layers next to the input layer of the IPNN.