The authors provide equivalent conditions for the commutator of two Toeplitz operators or two Hankel operators on the Hardy space to have finite rank. They also deal with a modified version of the problem studied in [X. Chen, K. Guo, K. Izuchi and D. Zheng, J. Reine Angew. Math. 578, 1–48 (2005; Zbl 1090.47017)] – namely, when is the product of two Hankel opertors equal to a finite-rank perturbation of a Hankel opertor? It is shown that \(H_f H_g - H_h\) has finite rank if and only if both \(H_fH_g\) and \(H_h\) have finite rank.