Mut'ah (temporary marriage) is a type of marriage in Islam, mentioned in Quran (Sura An Nisaa', verse 24).
Marriage contract (whether permanent or temporary) does not require witness, though it is recommended.
Mut'ah aims to save people from falling in sexual sinful acts.
Mut'ah should not be misused and should never be cause of hurting the feelings of your wife or children.
Mut'ah marriage can be only done with Muslim woman or a woman from Ahl al-Kitab, but in this case of Ahl Kitab, it needs the permission of your Muslim wife. This is according to the Fatwa of many leading scholars.