Disjoint cocircuits in matroids with large rank

We prove that, for any positive integers n, k and q, there exists an integer R such that, if M is a matroid with no M(Kn)- or U2,q+2-minor, then either M has a collection of k disjoint cocircuits or M has rank at most R. Applied to the class of cographic matroids, this result implies the edge-disjoint version of the Erdös–Pósa Theorem.