On Shabbos, many times my kids decide to wash after I have already cut lechem mishna. Are they are obligated to say hamotzi on lechem mishna? I once heard an eitzah of joining the two halves together again with toothpicks – does this satisfy the requirement of lechem mishna?
Children should are obligated to say the berachah over lechem mishnah — certainly for children over bar mitzvah, and for chinuch for those beneath bar mitzvah. If two halves look whole when joined together, this can be done, but it is not recommended. It is also hard to join them together in this way so that it looks whole, and depends on how the challah was cut. [Perhaps kids who wash late can be sanctioned by their having to do this themselves…]
Preferably, the kids should wash together with you. Alternatively, you could prepare a bunch of small whole chalos for them to make hamotzi on.
Sources: The idea of joining together two halves of a loaf is mentioned in Shulchan Aruch 168:2. However, Chasam Sofer (Orach Chaim 46) writes that this halachah is unclear, and should not be done lechatchilah.