  A $(d,h)$-decomposition of a graph $G$ is an ordered pair $(D, H)$ such that
$H$ is a subgraph of $G$ of maximum degree at most $h$ and $D$ is an acyclic
orientation of $G-E(H)$ with maximum out-degree at most $d$. In this paper, we
prove that for $l \in \{5, 6, 7, 8, 9\}$, every planar graph without $4$- and
$l$-cycles is $(2,1)$-decomposable. As a consequence, for every planar graph
$G$ without $4$- and $l$-cycles, there exists a matching $M$, such that $G - M$
is $3$-DP-colorable and has Alon-Tarsi number at most $3$. In particular, $G$
is $1$-defective $3$-DP-colorable, $1$-defective $3$-paintable and 1-defective
3-choosable. These strengthen the results in [Discrete Appl. Math. 157~(2)
(2009) 433--436] and [Discrete Math. 343 (2020) 111797].
