Can i claim a child on my taxes if he only lived with me for the first 6 months of the year?

Please explain.   We need more details.  And dates.

I had custody of my great grandson for a year, but he lived with me for only 6 months of the 2021calender yr so can I claim him for six months

There is no such thing as claiming a child for part of the year.   You either claim the child for the whole year or not at all.    That is why we want dates.   The person who had the child for the most nights in 2021 (at least 183 nights) is the custodial parent who can claim the child.