Is there constraint on reducibility while defining $\Sigma_i^p$?
In this definition, $M$ is constrained only to be a polytime relation, that is, you can compute $M$ in polynomial time. Note that there is no reduction here. The concept of reduction only comes up when you try to deduce that one problem is in a certain level of the hierarchy (or hard for a certain level of the hierarchy) given that another one is in some level of the hierarchy (hard for a certain level of the hierarchy).

For example, if $A \in \Sigma_i^P$ and $B$ polytime many-one reduces to $A$, then $B \in \Sigma_i^P$. If $A$ is $\Sigma_i^P$-hard and $A$ polytime many-one reduces to $B$ then $B$ is $\Sigma_i^P$-hard. If we consider reduction that are not polytime but are still in the polynomial hierarchy, we still get meaningful results that I encourage you to work out.