Sometimes people who are just coming off of bout of crying take what could be described as a stuttering inhalation. It's more easily observed with little kids than adults.
Is there a better word for that?
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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
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.
"Shuddering breath" is one description I have heard for the condition described as experiencing a few short involuntary breaths right at the end of taking a deeper inhalation breath. As stated above, a shuddering breath can often be observed towards the end of crying in children, or crying infants.
I have also heard this type of breathing referred to as triple breathing in western medicine, but I don't know if there is an explanation associated with this description.
In Traditional Chinese Medicine I believe there may also be a term for this type of involuntary shudder breathing for adults. It may be a symptom or a diagnostic indicator for something else that may be going on in the body.
CHOKING may be one of the answers. Refer choked
A breath holding spell (BHS) is a reflexive response that occurs in some healthy young children, usually between the ages of about eight months and two years. A typical breath holding spell lasts between two and 20 seconds. The child cries or gasps, forcibly exhales, stops breathing, and turns either blue (cyanotic form) or pale (pallid form).
Edit Wikipedia confirms that BHS is a recognized term, and for what it's worth I consider it a more accurate description for a "specific kind of inhalation after crying" than sobbing.
The second type are the Cyanotic breath-holding spells. They are usually precipitated by anger or frustration although they may occur after a painful experience. The child cries and has forced expiration sometimes leading to cyanosis (blue in color), loss of muscle tone, and loss of consciousness. The majority of children will regain consciousness. The child usually recovers within a minute or two, but some fall asleep for an hour or so. [...] Treatment The most important approach is to reassure the family, because witnessing a breath-holding spell is a frightening experience for observers. There is no definitive treatment available or needed for breath-holding spells, as the child will eventually outgrow them.