This question is closely related to What is the geometric object corresponding to a subalgebra in a polynomial ring. There, it is asked, given a subalgebra of an algebra $S \subset R$ over a field $k$ what is the corresponding geometric object, in the sense of algebraic geometry. The answer is that there are varieties $X,Y$ associated to $R,S$, respectively, along with a dense morphism $X \to Y$. Here the term "variety" is a bit looser than often used.

My question is the same, but for a specific type of subalgebra. Given a $k$-algebra $R$ and an ideal $I \subset R$, let $S = k + I \subset R$ be the $k$-subalgebra generated by the ideal $I$. What is the geometric interpretation of the variety $\mathrm{Spec}(S)$ and the morphism $\mathrm{Spec}(R) \to \mathrm{Spec}(S)$?

I am happy to assume that $R$ is a finite type $k$-algebra, and even that it is reduced. I am most concerned about the case when $R = k[x_1, ..., x_n]$ is just a polynomial ring.