Mut'ah marriage, literally "pleasure marriage", in Islam, refers to a practice that Muhammad apparently allowed in the early days of Islam, in which men were allowed to temporarily "marry" a woman, and then divorce her.
The primary sources for this practice are narrations such as:
Sahih al-Bukhari 4615
Narrated Abdullah:
We used to participate in the holy wars carried on by the Prophet (ﷺ) and we had no women (wives) with us. So we said (to the Prophet (ﷺ)). "Shall we castrate ourselves?" But the Prophet (ﷺ) forbade us to do that, and thenceforth he allowed us to marry a woman (temporarily) by giving her even a garment, and then he recited: "O you who believe! Do not make unlawful the good things which Allah has made lawful for you." (5:87)
Notice that the above, Muhammad appeals to a verse in the Quran to justify his allowance of "temporary marriage":
Surah 5:87
87 O believers! Do not forbid the good things which Allah has made lawful for you, and do not transgress. Indeed, Allah does not like transgressors.
Therefore, a sound case can be made that Muhammad not only allowed temporary marriage - which is prostitution in every sense, save in name only - but also called it a "good thing", as the above speaks of "the good things which Allah has made lawful for you".
Muhammad's endorsement of this practice is also recorded in Sahih al-Bukhari 5075, 5117-18, Sahih Muslim 1404a-c, 1405a-b, and Sunan Ibn Majah 1962.
Later, he went on to ban the practice, on the same day he banned the meat of domestic donkeys (Sahih al-Bukhari 5115, 5523, 6961, Sahih Muslim 1404b, 1405f, 1406a-l, 1407a-e):
Sahih al-Bukhari 4216
Narrated Ali bin Abi Talib:
On the day of Khaibar, Allah's Messenger (ﷺ) forbade the Mut'a (i.e. temporary marriage) and the eating of donkey-meat.
Or, according to some, it was allowed for Muhammad's entire life, and then Umar banned it (Sahih Muslim 1249, 1405c-e). Either way, the Sunni narrations are unanimous in stating that Muhammad allowed this practice during the early days of Islam.
Many Twelver Shia still practice Mut'ah, and it is mentioned repeatedly after the time of Muhammad and Ali in Kitab al-Kafi:
Kitab al-Kafi - Volume 5, Book 3, Chapter 94
4 Ali ibn Ibrahim has narrated from his father from Ibn Abi 'Umayr from 'Umar ibn 'Udhaynah from Zurarah who has said the following:
"Once 'Abd Allah ibn 'Umayr al-Laythiy came to abu Ja'far (a.s), and asked, 'What do you say about al-Mut'ah (advantageous marriage) of women?' He (the Imam) said, 'Allah has made it lawful in His book by the tongue of His Holy prophet, O Allah, grant compensation to Muhammad and his family worthy of their services to Your cause, thus it will remain lawful up to the Day of Judgment.' He then said, 'O abu Ja'far, how can someone like you say this, when 'Umar made it unlawful and prohibited it?' He (the Imam) said, 'Even if he has done so.' He said, 'I ask Allah to protect you against a thing that 'Umar has made unlawful.'
Kitab al-Kafi - Volume 5, Book 3, Chapter 104
1 Ali ibn Ibrahim has narrated from his father from Ibn Abi 'Umayr from certain persons of our people from Zurarah who has said the following: "I once asked abu Ja'far (a.s), saying, 'May I be sacrificed for you, a man marries a woman temporarily (through mut'ah) and its appointed time expires, then another man marries her until she becomes a stranger to him, then the first man marries her and she becomes stranger to him and this takes place three times and she marries three husbands; if it is permissible for the first man to marry her again.' He (the Imam) said, 'Yes, it is permissible, as many times as he wants; it is not like a free woman. She is hired and her case is similar to slave-girls.'"
In the scenario described above, a woman is essentially being treated as a prostitute, being exploited by multiple men in "temporary marriages".
Mut'ah marriage, which is a form of prostitution, is a mockery of marriage, and is prohibited in the Word of God (Leviticus 19:29, Deuteronomy 23:17, Matthew 5:32, Ephesians 5:5, etc.). By allowing this practice, Muhammad demonstrated himself not to be in the line of the Biblical prophets.