Irreversible adsorption of hard spheres at random site surfaces was studied theoretically. In contrast to the previous model of Jin [J. Phys. Chem. 97, 4256 (1993)] the dimension of the sites, having the shape of circular disks, was finite and comparable with the size of adsorbing spheres. Adsorption was assumed to occur if the sphere contacted the disk, i.e., when the projection of the sphere center was located somewhere within the disk only. Numerical simulation of the Monte Carlo type enabled one to determine the available surface function, adsorption kinetics, jamming coverage, and the structure of the particle monolayer as a function of the site density (coverage) and the size ratio particle/site, denoted by lambda. It was demonstrated that adsorption kinetics and the jamming coverage increased significantly, at a fixed site density, when the lambda parameter increased. It was also proven that the results derived from the Jin model were valid only if lambda>10.