  A spannning subgraph $F$ of $G$ is a $\{K_2,C_n\}$-factor if each component
of $F$ is either $K_{2}$ or $C_{n}$. A graph $G$ is called a
$(\{K_2,C_n\},n)$-factor critical avoidable graph if $G-X-e$ has a
$\{K_2,C_n\}$-factor for any $S\subseteq V(G)$ with $|X|=n$ and $e\in E(G-X)$.
In this paper, we first obtain a sufficient condition with regard to isolated
toughness of a graph $G$ such that $G$ is $\{K_2,C_{n}\}$-factor critical
avoidable. In addition, we give a sufficient condition with regard to tight
toughness and isolated toughness of a graph $G$ such that $G$ is
$\{K_2,C_{2i+1}|i \geqslant 2\}$-factor critical avoidable respectively.
