Lov{á}sz showed that a matching covered graph $G$ has an ear decomposition starting with an arbitrary edge of $G$. Let $G$ be a graph which has a perfect matching. We call $G$ cycle-nice if for each even cycle $C$ of $G$, $G-V(C)$ has a perfect matching. If $G$ is a cycle-nice matching covered graph, then $G$ has ear decompositions starting with an arbitrary even cycle of $G$. In this paper, we characterize cycle-nice claw-free plane graphs. We show that the only cycle-nice simple 3-connected claw-free plane graphs are $K_4$, $W_5$ and $\overline C_6$. Furthermore, every cycle-nice 2-connected claw-free plane graph can be obtained from a graph in the family ${\cal F}$ by a sequence of three types of operations, where ${\cal F}$ consists of even cycles, a diamond, $K_4$, and $\overline C_6$.