We prove that every cubic bridgeless graph $G$ contains a $2$-factor which intersects all (minimal) edge-cuts of size $3$ or $4$. This generalizes an earlier result of the authors, namely that such a $2$-factor exists provided that $G$ is planar. As a further extension, we show that every graph contains a cycle (a union of edge-disjoint circuits) that intersects all edge-cuts of size $3$ or $4$. Motivated by this result, we introduce the concept of a coverable set of integers and discuss a number of questions, some of which are related to classical problems of graph theory such as Tutte's $4$-flow conjecture or the Dominating circuit conjecture.