A Muslim man should conduct an Islamic marriage contract with a woman. Scholars differ as to whether someone can marry a Christian woman permanently. All allow temporary marriage and some scholars say that both temporary and permanent are allowed.
If however a Muslim man was ignorant and he for example read a marriage contract such as a civil one instead of Islamic (while he thought that this counts as marriage) then some scholars would say that the marriage wouldn’t be fornication.
This therefore could make the child one of shubha where they aren’t deemed as illegitimate.
If he found out his mistake though, then he must do an Islamic marriage as soon as possible.
Rgearding a Christian marriage contract, this should be asked from one's Marja and it should be stated whether the person was ignorant or not.
May Allah grant you success