The phrase "with sorrowful mien" generally refers to an expression or demeanor that is filled with sadness or grief. There are a number of synonyms that can be used to describe this state or feeling, including phrases like mournful countenance, pensive expression, or somber visage. Other similar terms might include words like wistful, melancholy, or forlorn, all of which suggest a sense of sadness or longing that can be seen in a person's face or mannerisms. Ultimately, the choice of words used to describe this state will depend on the mood and tone of the writing, as well as the specific context in which it is being used.