It is known that $A$-paths of length $0$ mod $m$ satisfy the Erdős-Pósa property if $m=2$ or $m=4$, but not if $m > 4$ is composite. We show that if $p$ is prime, then $A$-paths of length $0$ mod $p$ satisfy the Erdős-Pósa property. More generally, in the framework of undirected group-labelled graphs, we characterize the abelian groups $Γ$ and elements $\ell \in Γ$ for which the Erdős-Pósa property holds for $A$-paths of weight $\ell$.