The Kirkwood superposition approximation is investigated by making use of the expansion theorem for the potential of average force and its correction term is found in a form of the expansion in powers of particle number density ρ. The lowest order term of this correction is calculated for a special configuration of three particles in a fluid consisting of hard spheres and the Kirkwood approximation is shown to overestimate the distribution function of triplets in this case. The Kirkwood integral equation with correction is solved by expanding the radial distribution function in powers of ρ. The solution thus obtained is shown to be exact up to the order of ρ2, while the Kirkwood approximation itself does not yield the correct term of the order of ρ2. It is shown that the distribution function of triplets can, in principle, be expanded in terms of radial distribution functions and a few terms of this expansion are calculated explicitly, of which the first term just corresponds to the Kirkwood superposition approximation.