We prove that if $\mathcal{C}$ is a hereditary class of graphs that is polynomially $χ$-bounded, then the class of graphs that admit decompositions into pieces belonging to $\mathcal{C}$ along cuts of bounded rank is also polynomially $χ$-bounded. In particular, this implies that for every positive integer $k$, the class of graphs of cliquewidth at most $k$ is polynomially $χ$-bounded.