What's the word for a specific kind of inhalation after crying?

Solution 1:

Sobbing is how I would say it.

Edit: actually, M-W backs me up on this:

1 a : to catch the breath audibly in a spasmodic contraction of the throat

1 b : to cry or weep with convulsive catching of the breath

Solution 2:

I believe shuddering sob is the common term.

Long shuddering sobs were heard, cries, and deep sighs. Think how surprised everyone was when, on raising the sheets, they discovered Pinocchio half melted in tears!

A surprising number of references turn up when you Google the phrase.