will power of attorney free my mother from a court ordered nursing home

Is there a court order requiring your mother to stay in a nursing home?  I suppose a court order could provide for that but I've not seen that.  If there is, in fact, such a court order, then the answer to your question is "no", your power of attorney does not "trump" the court order.  However, you POA may give you the abiltiy to seek a modification of the order.  I would also wonder whether a guardian has been appointed for your mother.  If so, your power of attorney would not be valid against the decisions of the guardian.  Guardianship matters are very complex and I recommend that you seek the specific advice of an attorney.