What is for sure is that the child is for the father. However do you believe it is reasonable and practical that a child born from a temporary relationship ought to be left without a mother? Or the opposite, or being taken to an orphanage and being deprived of both a mother and a father? What kind of respect and prestige does a child born from a temporary relationship have in his/her society? Is it good for a child not to ever know his/her mother?! A child born from mut’ah (fixed-term marriage) is definitely an unintended outcome of a short-lived relationship! And this issue will remain with him his whole life! What would it feel like if we were to put ourselves in his place?
I very well know that in temporary marriage one can place the condition not to become pregnant beforehand but it is an undeniable reality that it might just occur. It might even happen regardless of both sides having agreed not to make a baby. So what is there to do if a woman, despite calculating her idda (waiting period), becomes pregnant? It isn’t merely about the guardianship, inheritance, nafaqa etc., it is a matter of how this child will conform to society with his/her predicament. What will happen with this child?
Temporary marriage is an easy and simple marriage for people that can’t practice permanent marriage, legitimized by God for a number of reasons, one of which is to prevent fornication and sexual corruption. Consequently, God has excluded a few of the rulings and conditions required in permanent marriage from it. Islam has determined what should happen to children born from temporary marriage dictating that they are no different from any other child in terms of nasab (lineage), inheritance, hidanah (guardianship), the right of nafaqa (sustenance), etc. and can by no means be deprived of the love and care of their real mother and father. It is possible that the culture in society today isn't prepared to accept such a marriage, but it must be normalized so that society isn't deprived of its benefits.
Of course, there are always exceptions for every law and rule but to oppose the entire idea for the sake of those few exceptions isn’t reasonable.
Before answering the question a few things must be pointed out:
First: We should always be able to clearly distinguish the purpose and aim of every divine ruling so as to expect the proper and intended outcome. For example we cannot condemn Jihad since it causes death, people are wounded and children are orphaned, when the purpose of jihad is to in fact defend and protect our religion, life, possessions and the ones we love from the harm of outsiders, so if a certain number are not willing to put their lives at stake for the sake of everybody else, everyone will obviously be in danger.
Second of all: The ahkam and Islamic rulings that are in some way social rulings should be analyzed concerning the vast majority of society, for there always is a chance that a general ruling won’t perfectly be compatible to each and every single person, these rare examples are the exceptions. Therefore the ahkam and laws pertaining to temporary marriage and its outcomes should be evaluated compared to the majority of the society, although there may be few exceptions as well.
Third of all: In a permanent marriage the hidanah (guardianship) of the child is the mother’s duty for two years, if a boy, and seven years if a girl, and subsequently the right of raising the child is given to the father afterwards.
With these points in mind we must say: temporary marriage is an easy and simple marriage for those who are incapable of practicing permanent marriage, which has been legitimized by Allah to prevent sexual corruption and bring about an honorable society. That’s why, in the case of temporary marriage, God Almighty has reduced the constraints of marriage.
One of the positive points of Islamic law is that in regard to temporary marriage Islam has planned out the proper thing to do in case a child is born; Islam declares that a child born from this type of marriage shouldn’t be treated any different from a child born from a permanent relationship. From the Islamic perspective he/she will be recognized as their children and they will be considered as his/her lawful mother and father, thus the child’s nafaqah (sustenance) will be the duty of the father and the child will inherit from both his mother and father. As a result the child will never feel ashamed and humiliated; because his/her mother and father have obeyed Allah by being in a relationship validated both by Islam and law instead of committing anything against the shariah (Islamic law). However we should be aware that the responsibility for taking care of the child is not only given to the father in temporary marriage but also in permanent marriage. In both cases, it is the mother’s right to take care of the child, for two years if it were a boy, and seven years if it were a girl, which she can give up to the father or use, but after that certain period this right will be passed on to the father. This hukm (ruling) might not be noticed as much in a permanent marriage due to the permanent relationship that holds the wife and husband together. Similarly in both types of marriage it is the father’s responsibility to provide all of his children with their financial needs and be their guardian after the period of hidanah. Obviously, God has arranged it this way in support of the mother, which is usually not as competent as the father in financially providing for the children and will most probably, in turn, have a hard time making ends meet. This is why Allah has placed the burden of nafaqah on the father’s shoulders. Although, if she wills the mother can accept this duty on her behalf for free or in exchange for payment, but no one can prevent her from seeing her child and showing him the love and affection he needs.
In addition to the above usually no one is looking to have a child from such a marriage. Either side can set the condition not to have a baby or if they do, allow the mother to raise the child for a period or forever. The truth is that not all conceptions that occur in mut’ah have been on accident; sometimes men whose first wives were not capable of having babies would do so just to have a child and would take care of that child in the best way possible.
It is possible not to witness many people nowadays in society that have admitted it gladly in their culture, but this tradition must be propagated and normalized throughout society with the aim of benefiting from its advantages. Of course, there are exceptions for every law and rule, so there might be a few examples of a child that has been born from a temporary marriage that doesn’t receive the amount of love and affection that he deserves but opposing the entire idea for the sake of a few exceptions isn’t reasonable.
 Tawdihul-Masa’el (annotated by Imam Khomeini), vol. 2, pg. 257, question 1504.
 Tabatabai, Mohammad Hosein, Al-Mizan fi Tafsir Al-Quran, vol. 15, pg. 15.
 Tawdihul-Masa’el (annotated by Imam Khomeini), vol. 2, pg. 486; Ayatullah Makarem Shirazi: Issue 2079, The iddah must be observed after the time for temporary marriage expires…and the children that are born by such marriage have all the rights that children born from permanent marriage have and inherit from their mother and father and relatives, although the man and woman don’t inherit from each other.
 Ibid; Imam Khomeini, Tahrir Al-Wasilah, vol. 2, pg. 312 (the chapter on birth and what is related to it), issues 12, 16 and 17.