(1)His main goal(主语)// was（系动词）//（表语） to carve out more time to spend with his wife and newborn child.

参考译文：他的主要目标是争取更多的时间和他的妻子和新生儿在一起。

(2)to carve out(不定式)// more time（宾语） to spend with his wife and newborn child（修饰宾语后置定语）

不定式短语语法结构分析如下

动词不定式：to carve out

不定式宾语：more time

修饰宾语后置定语：to spend with his wife and newborn child

(3)to carve out more time to spend with his wife and newborn child(动词不定式作后置定语,与被修饰名词“time”属于动宾关系）

=to carve out more time which he will spend with his wife and newborn child（定语从句作后置定语，关系代词which代替先行词time，并在定语从句作谓语动词spend宾语）

总结：to spend with是time后置定语并time是动宾关系。