Suppose that $M$ is a model of $\sf ZFC$, and we add some generic set $G$. Then it is not hard to see that for every $x\in M[G]$ it holds $M\subseteq M[x]\subseteq M[G]$.


Given $x\in M[G]$ such that $x\subseteq M$, can we find a forcing $P\in M[G]$ such that:


1. If $H$ is $P$-generic over $M[G]$ then $H\in M[G]$ (triviality).
2. If $H$ is $(P\cap M)$-generic over $M$, then $H\notin M$ (not so-triviality).
3. $x$ is $(P\cap M)$-generic over $M$ (relevance).


Obviously we're not talking about minimal forcing like Sacks and friends, but on forcing which have a lot of non-trivial intermediate forcings (like Cohen and friends).



Trivial forcings which are not very trivial
It is a standard result (e.g. Jech's Set Theory, lemma 15.43) that every intermediate model $W$ of ZFC with $M\subset W\subset M[G]$ has the form $W=M[G\cap\mathbb{C}]$ for some complete subalgebra $\mathbb{C}$ of the complete Boolean algebra $\mathbb{B}$ for the forcing giving rise to $G$. 


In particular, since $x\subset M$ we know that $M[x]\models \text{ZFC}$, and so in particular there is a partial order $P\in M$ which (we can easily arrange) is nontrivial and adds $x$ via $G\cap P$. This achieves nontriviality and relevance (2 and 3). 


Now, my observation is that your triviality condition 1 is also easy to obtain by modifying $P$, to add a single new atom below every element of $P$, but with a node that is not in $M$. For example, in $M[G]$, let $P^+$ be the poset $P$ together with a new atom, $G$, considered as a single point below every element of $P$. So $P^+$ is atomic, and hence trivial, but $P^+\cap M=P$, which satisfies your requirements in 2 and 3. 


So the answer is yes.


