Lay­ered $H$-par­ti­tions of graphs (with small lay­ered width, in which $H$ has small treewidth) are a re­cently-in­tro­duced tool that have been used to solve long­stand­ing prob­lems on queue lay­outs, non-repet­i­tive colour­ing, and 3-d graph draw­ing. Such par­ti­tions are known to exist for pla­nar graphs and, more gen­er­ally, bounded genus graphs. In the cur­rent paper, we prove that every $k$-pla­nar graph has a lay­ered $H$-par­ti­tion of lay­ered width $O(k^2)$ in which $H$ has treewidth $O(k^3)$. This is the first re­sult of this type for a non-minor-closed class of graphs and im­plies that $k$-pla­nar graphs have both queue num­ber and non-repet­i­tive chro­matic num­ber upper-bounded by a func­tion of $k$. The for­mer re­sult was pre­vi­ously shown using a com­bi­na­tion of $H$-par­ti­tions for pla­nar graphs with \tex­tit{ad hoc} meth­ods. The lat­ter re­sult is new, for all $k\ge 1$. These re­sults ex­tend to $(g,k)$-pla­nar graphs, the nat­ural gen­er­al­iza­tion of $k$-pla­nar graphs to to genus-$g$ sur­faces (rather than the genus-$0$ plane).