Can she ask for a divorce if her husband takes a second wife?

Praise
be to Allah

Firstly:

A woman does not have the power to divorce (talaaq) her
husband; rather divorce is in the hand of the man, because the Prophet
(blessings and peace of Allah be upon him) said: “(The right of divorce)
belongs to the one who takes hold of the calf [i.e., her husband].” Classed
as hasan by al-Albaani in Saheeh Ibn Maajah 

The calf referred to here is the calf of the wife; this is a
metaphor for intercourse. Hence the one who has the right of divorce (talaaq)
is the one who has the right of intercourse. 

The Standing Committee was asked: If a woman divorces (talaaq)
her husband, does she have to offer expiation, and what is that expiation?

They replied: If a woman divorces her husband, it does not
count as such and she does not have to offer any expiation, but she should
ask Allah for forgiveness and repent to Him, because her issuing a divorce
to her husband is contrary to the shar‘i evidence, which indicates that
divorce is in the hand of the husband or whoever is acting on his behalf
according to sharee‘ah.

End quote from Fataawa al-Lajnah ad-Daa’imah (vol. 1 –
20/11). 

For more information, please see the answer to question no.
111881

Secondly: 

The husband is not obliged to seek the permission or approval
of his wife if he wants to take a second wife, but part of good treatment is
to console her by taking measures to reduce the distress that women
naturally feel in such situations. For more information, please see the
answer to question no. 12544.

Our advice to the woman whose husband takes a second wife is
to be patient, seek reward with Allah and accept what Allah has decreed for
her. No one knows where goodness lies. Allah, may He be exalted, says
(interpretation of the meaning):

“…it may be
that you dislike a thing which is good for you and that you like a thing
which is bad for you. Allah knows but you do not know”

[al-Baqarah 2:216]. 

But if it so happens that a woman cannot put up with that and
the fears that she will not be able to fulfil her husband’s rights over her
if he takes a second wife, then she may request khula‘ and return the mahr
to the husband, because of the report narrated by al-Bukhaari (48677) from
Ibn ‘Abbaas (may Allah be pleased with him), according to which the wife of
Thaabit ibn Qays came to the Prophet (blessings and peace of Allah be upon
him)  “O Messenger of Allah, I do not find any fault with Thaabit ibn Qays
in his character or his religious commitment, but I do not want to commit
any act of kufr after becoming a Muslim.” The Messenger of Allah (peace and
blessings of Allah be upon him) said to her, “Will you give back his
garden?” Because he had given her a garden as her mahr. She said, “Yes.” The
Prophet (peace and blessings of Allah be upon him) said to Thaabit: “Take
back your garden, and divorce her.”. 

Ibn Hajar (may Allah have mercy on him) said: The words “but
I do not want to commit any act of kufr after becoming a Muslim” mean: If I
stay with him, I would hate to fall into anything that constitutes kufr. It
is as if she was indicating that her intense dislike of him might prompt her
to make an outward display of kufr so that her marriage to him would be
annulled. She knew that that was haraam, but she feared that intense hatred
might push her to do that. Or it may be that what she meant by kufr was
ingratitude to her husband, which refers to the wife falling short in
fulfilling the rights of the husband.

End quote from Fath al-Baari (9/399). 

But if the husband is not well off and cannot afford to spend
on his wife’s maintenance and do what he is obliged to do of taking care of
her, then she may refer the matter to the Islamic judge (qaadi) and request
an annulment of the marriage. 

For more information, please see the answers to questions no.
452, 169847 and
133859 

And Allah knows best.