Let ĉ_n be the nth Cauchy number of the second kind, defined by the generating function x/(1 + x) ln(1 + x) = Σ^∞_n=0 ĉ_nxⁿ/n!. We obtain explicit expressions for (ĉ_l + ĉ_m)ⁿ := Σⁿ_j=0 (ⁿ_j)ĉ_l+jĉ_m+n−j with arbitrary fixed integers l, m ≥ 0. The corresponding expression for Bernoulli numbers has been studied by Agoh and Dilcher, and that for Cauchy numbers of the first kind by the author.