Perfectly diffuse light is a fundamental assumption in the Kubelka-Munk (KM) model. This assumption is, however, seldom fulfilled by real media. In this work, we build a bridge between a weakly absorbing medium with an imperfectly diffuse light distribution and the corresponding KM model relying on a perfectly diffuse light distribution. We multiply the apparent K and S phenomenal coefficients deduced from a learning set medium by a scaling factor expressing the ratio of light diffuseness between the target medium and the learning set medium. For the target medium, thanks to this diffuseness scaling factor, theoretically predicted reflection and transmission spectra agree with the corresponding measured spectra. The illumination geometry, the optical properties and the thickness of the medium have an impact on the light diffuseness and therefore on the proposed diffuseness scaling factor.