Suppose $M$ is a countable transitive model for ZFC. Suppose $X\subset M\cap \textbf{ON}$ is bounded in $M\cap \textbf{ON}$. Why does $M[X]$ (the least ctm $N$ such that $M \subset N$ and $X \in N$) exists? Furthermore, is it true that there exists a forcing notion $P \in M$ and a $P$-generic filter over $M$, $G$, such that $M[G]=M[X]$?

I have looked for this information in some books such as Kunen, Jech and Kanamori's but I have found nothing.

Edit: @Elliot Glazer's answer shows that $M[X]$ is not always a generic extension. So I am changing the question a bit.

Suppose that there exists a forcing notion $Q$ and a generic filter $H$ over $M$ such that $X \in M[H]$ is a subset of $M\cap \mathbf{ON}=M[H]\cap \mathbf{ON}$. Is it true that there exists a is a forcing notion $P \in M$ and a $P$-generic filter over $M$, $G$, such that $M[G]=M[X]$?