They were finally able to have the child they had longed for so intensely.

intensely 是形容词 intense 的副词形式，由 intense（强烈的）加上副词后缀 -ly 构成，意思是"强烈地、极度地"。 在这里修饰动词 longed（渴望），表示渴望的程度非常强烈。