It means "to cry and sniffle" , that's why it's "鼻子" :)

对于这个词的由来不是很清楚，只知道是一个习惯说法。哭的时候，会留鼻涕，感觉鼻子酸酸的，可能就是哭鼻子的由来吧。这个词，通常是跟小孩子说。因为说这个词的时候，总会带有撒娇的语气

It's originally used for kids. When they cry, usually they have running nose. So,, And also, 哭鼻子 is a little girly, so I don't think it's good to use it in your daily life.

这是一个短语。

这是俗语 就是哭了的意思