Magnetic phases and phase transitions in artificial spin ice, constructed from tailor-made arrays of nanomagnets, are currently subjects of great interest. The authors present measurements of the diffuse soft x-ray resonant magnetic scattering in artificial spin ice, where the moments of the magnets, arranged on the sites of the kagome lattice, are highly dynamic. Comparing experimental scattering patterns with the patterns calculated from Monte Carlo simulations based on a needle-dipole model, they show the emergence of quasi-pinch-points in the kagome ice I phase, and explain their relation to the pinch-point singularities in spin ice pyrochlores. As in the bulk pyrochlore spin ices, measurement of diffuse scattering from artificial kagome spin ice provides unique information on the magnetic correlations and can be applied to a number of other problems concerning nanomagnetic systems.