 Solution

Let $c=\bigcup G$ and $d(n)=c(2n)$. DefineThis is a filter: restrictions of finite pieces of $d$ remain in $H$, and the union of two members is a common stronger condition.

To prove genericity, take a dense set $D\in M$. Let $E_D$ consist of conditions $q\in \mathrm{Fn}(\omega ,2)$ for which some $p\in D$ satisfiesThe set $E_D$ is dense. Indeed, given $q$, first read its finitely many assigned even coordinates as a condition $p_0$ on $\omega$. Choose $p\le p_0$ in $D$, and extend $q$ by setting $q(2n)=p(n)$ at the remaining coordinates of $p$. Since $E_D\in M$, the generic filter $G$ meets it. For $q\in G\cap E_D$, the corresponding $p\in D$ is a finite subfunction of $d$, so $p\in H\cap D$.

Thus $H$ meets every dense subset belonging to $M$. Moreover, each singleton $\{(n,d(n))\}$ belongs to $H$, and hence