Praise be to Allah.
Honouring the father’s wife, treating her kindly and asking after her are part of honouring one’s father and upholding ties of kinship with him, especially since he is asking you to do that and urging and encouraging you to do so.
The father’s wife is not one of the relatives (rahm) with whom one is obliged to uphold ties of kinship, unless she is related to you by blood, but not asking after her is regarded as offending the father, and falling short in upholding ties with him.
It says in Subul al-Salaam (2/628): The scholars differed as to the definition of the relatives with whom one is obliged to uphold ties of kinship. It was said that they are the relatives to whom marriage is forbidden. Based on this, the children of paternal and maternal uncles (i.e., cousins) are not included in this. This who hold this view quote as evidence the fact that it is haraam to be married to a woman and her paternal aunt or maternal aunt at the same time, because that would result in the severing of family ties.
And it was said that it applies to anyone who is connected by inheritance, which is indicated by the words of the Prophet (peace and blessings of Allaah be upon him): “Then the closest to you and the next closest.”
And it was said that it applies to anyone to whom you are related, whether you would inherit from him or not.
Moreover, as al-Qaadi ‘Iyaad said: there are degrees of upholding the ties of kinship, some of which are higher than others. The least of them is not forsaking them and upholding ties by speaking, even if it is only saying salaam. That varies according to ability and need, and some of it is obligatory and some is mustahabb. If a person upholds the ties of kinship to some extent but not fully he is not regarded as having severed the ties of kinship, and if he falls short of what he is able to do and should do, he is not regarded as upholding the ties of kinship.
Al-Qurtubi said: The ties of kinship which should be upheld are general and specific. The general ties are the ties of religion which must be upheld with friendship, sincerity, justice, fairness and fulfilling obligatory and mustahabb rights.
Specific ties include, in addition to that, spending on relatives, checking on them and forgiving their mistakes. End quote.
Conclusion: The father’s wife should be honoured and ties with her upheld, and you should check on her and ask after her, based on ties of religion and because upholding ties with her is part of upholding ties with your father.
And Allaah knows best.