Consider a matroid $M$ on a ground set $E$ with a subset $X \subseteq E$ such that $X$ is both a circuit and a hyperplane of $M$. If $\mathcal B(M)$ is the set of bases of $M$, then we can form a new matroid by the so-called relaxation technique, that is, a new matroid with set of bases $\mathcal B(M) \cup \{X\}$.
Question: Is there an extensive paper that deals with the properties of these matroid relaxations?
These are often called circuit-hyperplane relaxations. A relatively recent paper using it is Matroid fragility and relaxations of circuit hyperplanes, where the authors, Jim Geelen and Florian Hoersch, talk about its role with respect to $\mathbb{F}$-representability and Rota's conjecture.