Yes, in most places you can marry your second cousin. However, some people and some cultures are uncomfortable with such a marriage.
All the States in the US allow the marriage of 2nd cousins. A number of states allow the marriage of 1st cousins.
From religious point of view, most religions do not forbid it. Christianity, Judaism and Islam say nothing about the prohibition of first cousins marrying let alone second cousins.
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Yes, it is legal.
Yes yes can marry your 6th cousin but many societies disapprove of marrying your first cousin
no
Your mother's first cousin is YOUR first cousin, once removed. The child of your mother's first cousin is YOUR second cousin.
Whether you can marry your first cousin twice removed depends in part on the laws of the place where you propose to have the marriage performed. In most of the US, you can marry your first cousin, so marrying your first cousin twice removed would also be permitted.