  The planar Tur\'an number of a graph $H$, denoted by
$ex_{_\mathcal{P}}(n,H)$, is the largest number of edges in a planar graph on
$n $ vertices without containing $H$ as a subgraph. In this paper, we continue
to study the topic of "extremal" planar graphs initiated by Dowden [J. Graph
Theory 83 (2016) 213--230]. We first obtain an improved lower bound for
$ex_{_\mathcal{P}}(n,C_k)$ for all $k\ge 13$ and $n\ge
5(k-6+\lfloor{(k-1)}/2\rfloor)(k-1)/2$; the construction for each $k$ and $n$
provides a simpler counterexample to a conjecture of Ghosh, Gy\H{o}ri, Martin,
Paulos and Xiao [arxiv:2004.14094v1], which has recently been disproved by
Cranston, Lidick\'y, Liu and Shantanam [Electron. J. Combin. 29(3) (2022)
\#P3.31] for every $k\ge 11$ and $n$ sufficiently large (as a function of $k$).
We then prove that $ex_{_\mathcal{P}}(n,H^+)=ex_{_\mathcal{P}}(n,H)$ for all
$k\ge 5$ and $n\ge |H|+1$, where $H\in\{C_k, 2C_k\}$ and $H^+$ is obtained from
$H$ by adding a pendant edge to a vertex of degree two.
