Final answer:
The correct term for the woman in question is a concubine, which refers to a woman living with a man without being legally married to him and often being of lower status than a wife.
Step-by-step explanation:
The passage suggests that the woman was associated with a man of status and was referred to by his mother in a way that diminished her proper station. Given the context and the roles described, the correct term for the woman who cohabits with an important man, particularly if she is a duke's concubine and the mother of his heir, would be concubine.
A concubine is a woman who lives with a man to whom she is not legally married and is typically of a status below that of a wife. Although she may have a recognized position within society and hold certain rights, her status is not equivalent to that of a legally married spouse, such as a duchess or queen, and she is not simply a mistress, who might not have any official recognition.