  The size-Ramsey number $\hat{R}(\mathcal{F},H)$ of a family of graphs
$\mathcal{F}$ and a graph $H$ is the smallest integer $m$ such that there
exists a graph $G$ on $m$ edges with the property that any colouring of the
edges of $G$ with two colours, say, red and blue, yields a red copy of a graph
from $\mathcal{F}$ or a blue copy of $H$. In this paper we first focus on
$\mathcal{F} = \mathcal{C}_{\le cn}$, where $\mathcal{C}_{\le cn}$ is the
family of cycles of length at most $cn$, and $H = P_n$. In particular, we show
that $2.00365 n \le \hat{R}(\mathcal{C}_{\le n},P_n) \le 31n$. Using similar
techniques, we also managed to analyze $\hat{R}(C_n,P_n)$, which was
investigated before but only using the regularity method.
