Find the set of all edges which there is a cycle such that for every $e' \in C, e'\neq e$ : $w(e')\leq w(e)$
Suppose $E_{i}$ be the set of all edges that have weight $i$. Now, we want to check if any of these edges are part of the solution or not. Note that, we can simply delete all the edges from the graph of the weight strictly greater than $i$ since they should not come in a cycle containing $e \in E_{i}$. Let $G_{i}$ be the new graph after deletion of these edges. Now, we just have to solve the following new problem:
Updated Problem: Find all the edges in $E_{i}$ that form a cycle in $G_{i}$
Try solving this new problem on your own. In case, you need any help, check this link for the solution to this problem in $O(|V|+|E|)$ time.
Since there are only $10$ such graphs $G_{i}$. The overall time would remain $O(|E|+|V|)$.