We propose two solvable cluster growth models, in which an irreversible aggregation spontaneously occurs between any two clusters of the same species; meanwhile, monomer birth or death of species A occurs with the help of species B. The system with the size-dependent monomer birth/death rate keel K(i,j) = Jijv is then investigated by means of the mean-field rate equation. The results show that the kinetic scaling behaviour of species A depends crucially on the value of the index v. For the model with catalysis-driven monomer birth, the cluster-mass distribution of species A obeys the conventional scaling law in the v ≤ 0 case, while it satisfies a generalized scaling form in the v＞0 case; moreover, the total mass of species A is a nonzero value in the v＜ 0 case while it grows continuously with time in the v＞0 case. For the model with catalysis-driven monomer death, the cluster-mass distribution also approaches the conventional scaling form in the v ＜ 0 case, while the conventional scaling description of the system breaks down in the v ≥ 0 case. Additionally, the total mass of species A retains a nonzero quantity in the v ＜0 case, but it decreases to zero with time in the v ≥ 0 case.