The structure of matroids with a spanning clique or projective geometry

Let s,n≥2 be integers. We give a qualitative structural description of every matroid M that is spanned by a frame matroid of a complete graph and has no Us,2s-minor and no rank-n projective geometry minor, showing that every such matroid is 'close' to a frame matroid. We also give a similar description of every matroid M with a spanning projective geometry over a field GF(q) as a restriction and with no Us,2s-minor and no PG(n,q′)-minor for any q′>q, showing that such an M is 'close' to a GF(q)-representable matroid.