If not a Mulberry of some sort, then I'm not able to help further.
It looks like a woody plant that has at least a year's growth. There's a main stem, and several side branches - one of which has the new leaves you have shown. That looks like a couple or three season's worth of growing - but since you are in mild CA, maybe that happens in one year.
Did this plant sprout in the pot shown? Or was it transplanted into it from somewhere else?