In the modern context one cannot answer if a person is allowed to leave an abusive and unrighteous mother, because women are working these days and can make a living or subsistence on their own.
However, the ancient Hinduism's dharmika scriptural view on women is that they shouldn't work at all, and must always be used the care, guidance and guardianship of a male relative throughout her lifetime (i.e., under her father's control when a child, under her husband's control when married, and under her sons or other close male relatives if both father and husband are not present), as I discuss in this answer.
Now, coming to the importance of Parents and specifically the Mother.
Consider these excerpts from the two Mahā-Purāṇas.
25-26. Just as Maheśvara, the Lord of Devas, is the most excellent one
among Suras, just as Gaṅgā is the greatest of all rivers, just as
Brāhmaṇa is the greatest of all castes, just as Oṃkāra is the foremost
among all Akṣaras (syllables), just as mother is the greatest among
those worthy of adoration, just as father is the greatest (most
respectable) of all elders,...
For life’s sake he should never talk hatefully with the elders. Even though excellent in other virtues, a man who hates his elders, falls
35b-40a. Out of these, five are to be specially respected: Out of
these the first three are the greatest, and out of these (three) the
mother is the most venerable. He who desires his welfare, should
specially respect, with all efforts, even by casting his life, these
*As long as the father and the mother remain unchanged (in mind or attitude), a son should, giving up everything (else), be attached to
them. If, by the good qualities of their son, the father and the
mother are well pleased, then, the son would obtain all religious
40b-45a. There is no (other) deity like the mother, and no (other)
venerable person like the father. There can never be in their case
doing good to them in return. He should always do what pleases them
by means of deeds, thought and words. Without being permitted by
them, he should not follow any other course of conduct except (in the
case of rites having) liberation as their fruit or the obligatory and
In this world, a man attains greatness by serving his mother.
Thus, a Mother is the greatest person in an individual life and she must be respected despite her any sort of behavior, as per the scriptures.
Some ideal sons in the Scriptures -
Now let me provide some examples from the scriptures, about sons who respected and catered to their parents despite all sorts of hardships.
The foremost example is of God Sri Râma. It's needless to say what all Kaikeyi did to the Lord, and even her husband. Even so then the Lord set an exemplar example of an ideal son. He even sought forgiveness from his father on Kaikeyi's behalf, as found in the Valmiki Ramayan Verse 7-119-25-27.
In a story found in the Padma Maha-Purāṇa Verse 2.4.15-45 a pious Brahmin Śivaśarman's son named Somaśarman, kept up with his parents despite several abuses and hardships.
To Conclude -
A mother is the greatest person for an individual who must be respected, come what may. Even if she's abusive or rude, it's the foremost duty of the son to take care of her, and support her, notwithstanding her unrighteousness, since women are not allowed to work outside home, as per scriptures. The two exemplar examples of ideal sons are a testimony to such ideals and guidelines for a son.
So, yes even if the Mother is not right and causes trouble, we should learn to adjust with her. That's correct.