There are a lot of different meanings for t"Hitch" but here it means catch/jerky

a hitch here is a catch with the breath stuck in the throat like there's a blockage. It means the person is gulping in air or gasping for breath at the same time as sobbing.

It may sound like a hiccup or rasping sobs

the chest may also be heaving up and down with the effort to breathe

some examples used in literature

https://landwords.wordpress.com/2019/09/20/the-lady-at-the-f...

"As her face distorted into one of pure despair, she lets out a hitched sob, chest heaving from the emptiness she now holds inside..."

https://www.wattpad.com/724807950-poignant-poems-silent-tear...
...A hitched sob stuck in his throat

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Note added at 6 hrs (2020-09-08 15:05:36 GMT)
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https://forum.wordreference.com/threads/i-hitched-gusts-of-a... England, English

I agree that it cannot mean "fasten". It seems to mean to take in gulps of air convulsively as one does when sobbing. It's almost the definition of sob: "To catch the breath in a convulsive manner"

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Note added at 1 day 3 hrs (2020-09-09 11:46:10 GMT) Post-grading
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Glad to have helped. Just going to edit glossary to remove one of those bs. from sobbbing - to sobbing. Sorry!