It could be argued he (edit James I/VI) was the closest thing to a Plantagenet claim.

The legitimate Plantagenet line died with the execution of Edward, Earl of Warwick, in 1499. While some may say the line continues today, it requires going through 2 illegitimate lines, or by assuming Edward IV was illegitimate, or his marriage and therefore children were illegitimate, which would make the line continue with the descendants of his brother George. I don't know of any claims pressed by those descendants.

Elizabeth I's successor, James I of England/James VI of Scotland descended from Margaret Tudor, Henry VIII's sister. Therefore he came from 2 cadet branches of the extinct Plantagenets: Lancaster and York. Margaret Tudor's father was Henry VII, the first Tudor monarch, whose mother descended from John of Gaunt, a Plantagenet who created the House of Lancaster. Margaret Tudor's mother was Elizabeth of York, who descended from Edmund of Langley, a Plantagenet who created the House of York.

With the legitimate Plantagenet line being extinct, James I/VI descended from both cadet branches. He was an only child.