Origami structures demonstrate great theoretical potential for creating metamaterials with exotic properties. However, there is a lack of understanding of how imperfections influence the mechanical behaviour of origami-based metamaterials, which, in practice, are inevitable. For conventional materials, imperfection plays a profound role in shaping their behaviour. Thus, this paper investigates the influence of small random geometric imperfections on the nonlinear compressive response of the representative Miura-ori, which serves as the basic pattern for many metamaterial designs. Experiments and numerical simulations are used to demonstrate quantitatively how geometric imperfections hinder the foldability of the Miura-ori, but on the other hand, increase its compressive stiffness. This leads to the discovery that the residual of an origami foldability constraint, given by the Kawasaki theorem, correlates with the increase of stiffness of imperfect origami-based metamaterials. This observation might be generalizable to other flat-foldable patterns, in which we address deviations from the zero residual of the perfect pattern; and to non-flat-foldable patterns, in which we would address deviations from a finite residual.