Just some further thoughts: we must take into account the cumulative effect of suffering depression, isolation and marriage breakdown and not having a stable place to live as being heavy burdens on any young mother. They proved too much for this young mother.
But as we focus on understanding the mother's motive, we need to also focus that the child had a right to life and care. Her murder should not be dismissed on the above grounds alone.
We have to ask who were the decision makers who allowed this child to remain in the care of such an ill mother?
Retrospectively, it appears that there was an unresolved mental health issue here, but also pressing emotional/social circumstances [i.e. marriage breakdown, family conflict, isolation, lack of stable accommodation, lack of parental support and lack of stable supports] which should have been a 'red flag' to social welfare service decision makers who have a duty to protect the rights of the child.
IMO Questions need to be asked of the health & welfare services involved in this case. Did they get caught up in the heat of parental/familial conflict/separation and fail to make an assessment of this young mother's mental/emotional/social ability to care for this child?