Does Hungarian use "hát" as a short colloquial term for "people who support you"?
No, I don't think so. Does English? Can you give an example? Maybe there is some Hungarian equivalent.
I know of "I have your back", but that is something else.
I was actually thinking about Polish, where "plecy" can also mean "people you know who would support you and could get you out of trouble". You can have "duże plecy", or many such people, or "nie mieć pleców", or not know any people who matter, and so on.
I would translate that to either
"to support" - "támogatni"
"to verify" - "igazolni".
I don't recall any single Hungarian word that would cover both of those meanings.
Maybe. There are a few meanings of "backup". One is help or support, reinforcement. For those, I would use "erősítés", "tartalék". Etc.
Possibly at one's back; or covering one's back. In rugby a "back" is a lage defnse player.