Given $M$ a wellfounded transitive set model of ZFC, and $x$ a set which is not in $M$, is there always a 'smallest wellfounded transitive model' $M[x]$ of ZFC which extends $M$ and contains $x$?

I believe the answer is NO, because there might not be a 'canonical choice-function' for the set $x$ which we want to add. But I'm not sure how to make that precise.

If the above idea can be made precise and the answer is in fact NO, then the natural follow-up question would be: What if we only demand $M[x]$ to be a model of ZF?