We report Gemini-North Gemini Multi-Object Spectrograph observations of the inflated hot Jupiter HAT-P-32b during two primary transits. We simultaneously observed two comparison stars and used differential spectrophotometry to produce multiwavelength light curves. ‘White’ light curves and 29 ‘spectral’ light curves were extracted for each transit and analysed to refine the system parameters and produce transmission spectra from 520 to 930 nm in ≈14 nm bins. The light curves contain time-varying white noise as well as time-correlated noise, and we used a Gaussian process model to fit this complex noise model. Common mode corrections derived from the white light-curve fits were applied to the spectral light curves which significantly improved our precision, reaching typical uncertainties in the transit depth of ∼2 × 10⁻⁴, corresponding to about half a pressure scale height. The low-resolution transmission spectra are consistent with a featureless model, and we can confidently rule out broad features larger than about one scale height. The absence of Na/K wings or prominent TiO/VO features is most easily explained by grey absorption from clouds in the upper atmosphere, masking the spectral features. However, we cannot confidently rule out clear atmosphere models with low abundances (∼10⁻³ solar) of TiO, VO or even metal hydrides masking the Na and K wings. A smaller scale height or ionization could also contribute to muted spectral features, but alone are unable to account for the absence of features reported here.