Hey welcome to the board!

No...I don't think Emma has one of Jean's pieces. At the end of Here Comes Tomorrow its pretty evident that Jean pushes Scott to be with Emma to avoid the horrible future.

Also in Phoenix Endsong, they never alluded to a piece of Jean being inside of Emma. Infact, the only direct line Jean has with Emma is "Now get out of that stupid body" and she rips out the Phoenix from her body (with no evidence that a piece stayed in her).

So no...I don't think so based on that. But if you have ideas or evidence that proves other wise...tell me!