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Optical micro-spectroscopy is an invaluable tool for studying and characterizing samples ranging from classical semiconductors to low-dimensional materials and heterostructures. To date, most implementations are based on point-scanning techniques, which are flexible and reliable but slow. Here, we describe a setup for highly parallel acquisition of hyperspectral reflection and photoluminescence microscope images using a push-broom technique. Spatial as well as spectral distortions are characterized and their digital corrections are presented. We demonstrate close-to diffraction-limited spatial imaging performance and a spectral resolution limited by the spectrograph. The capabilities of the setup are demonstrated by recording a hyperspectral photoluminescence map of a CVD-grown MoSe$_2$-WSe$_2$ lateral heterostructure, from which we extract the luminescence energies, intensities and peak widths across the interface.