The popular Longuet-Higgins model for random ocean waves consists of a number of frequency components involving random phases. The model follows the Gaussian distribution in the limit of an increasing number of components, but deviates from Gaussianity in the pre-limit. This deviation is quantified in this work for the autocovariance function of the squares of the Longuet-Higgins model, and is substantial even for the moderately large numbers of frequency components typically used in practice. This is shown to have important implications on the uncertainty quantification for the Longuet-Higgins model, when constructing confidence intervals for quantities involving squares of the model for one long record and many shorter records.