Hadwiger number under matching contraction
There is no such function - here is an example where $h(G)$ is arbitrarily large while $h(G/M) \leq 4$. Make $G$ by taking two copies of an $n \times n$ grid and making every vertex of one grid adjacent to the corresponding vertex of the other grid.

$G$ contains a clique of size $n$ as a minor (i.e $h(G) \geq n$). In particular the $i$'th vertex of the clique is obtained by contracting the $i$'th row of the first grid and the $i$'th column of the second grid and then contracting these two vertices together.

Let $M$ be the matching that connects each vertex of the first grid with its copy in the second grid. $G/M$ is an $n \times n$ grid, so $G/M$ is planar and hence $h(G/M) \leq 4$.