My cousin has just had a baby.

has 是动词 have 的第三人称单数现在时形式，表示"拥有"或"经历"。 had 是动词 have 的过去分词形式，在这里与 has 构成现在完成时态，表示"已经完成"的动作。 has had 构成现在完成时，强调"刚刚完成"生孩子的动作。