The phrase "doesn't have much" is correct and usable in written English.
You can use it when you want to express that something has a minimal amount of something. For example, "He doesn't have much time before his meeting."
Idiom
"Don't have a cow ".
Don't overreact.
Exact(58)
Wales doesn't have much choice.
He doesn't have much time.
It doesn't have much to say.
"It doesn't have much commercial value.
Tsipras doesn't have much choice, however.
She doesn't have much choice.
He doesn't have much equipment.
"It doesn't have much structure.
It just doesn't have much political traction.
Frank doesn't have much sympathy for Wally.
Rwanda doesn't have much in natural resources.
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Since I tried Ludwig back in 2017, I have been constantly using it in both editing and translation. Ever since, I suggest it to my translators at ProSciEditing.

Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com