We reconsider the inference of spatial power spectra from angularclustering data and show how to include correlations in both the angularcorrelation function and the spatial power spectrum. Inclusion of thefull covariance matrices loosens the constraints on large-scalestructure inferred from the Automated Plate Measuring (APM) survey byover a factor of 2. We present a new inversion technique based onsingular-value decomposition that allows one to propagate the covariancematrix on the angular correlation function through to that of thespatial power spectrum and to reconstruct smooth power spectra withoutunderestimating the errors. Within a parameter space of the cold darkmatter (CDM) shape Γ and the amplitude σ_{8}, wefind that the angular correlations in the APM survey constrain Γ