Without any condition on the matroid structure, there is really no reason for your inequality to hold.

For example, take $X=\{a\},Y=\{b\}$ so that $X+Y=\{a+b\}$ where $a,b$ are any nonzero elements of $G$. Then take any matroid structure such that $a+b$ is a loop but $a,b$ and $0$ are not. We have $0=r(X+Y)<r(X)+r(Y)-r(H)=1$.

Maybe you want to add some kind of compatibility condition between the group and matroid structures to make your question more interesting?

Without any condition on the matroid structure, there is really no reason for your inequality to hold.

For example, take $X=\{a\},Y=\{b\}$ so that $X+Y=\{a+b\}$ where $a,b$ are any nonzero elements of $G$. Then take any matroid structure such that $a+b$ is a loop but $a,b$ and $0$ are not. We have $0=r(X+Y)<r(X)+r(Y)-r(H)=1$.

Maybe you want to add some kind of compatibility condition between the group and matroid structures to make your question more interesting?

EDIT: Even in a very simple and and well-behaved setting (say $G$ cyclic and $M$ uniform), the inequality fails. For example, let $G=\mathbb{Z}/N\mathbb{Z}$ with the $n$-uniform matroid structure, where $n<N/2$. Let $X\subset [0, N/2)$ be any subset of size $n$. Then $r(X+X)=r(X)=r(Y)=n$ and $H$ is trivial so $r(H)=1$. Hence your inequality fails whenever $n>1$.