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"slight majority" is correct and can be used in written English.
You can use it to describe a small numerical majority. For example, "The slight majority of 34 to 32 won the election for the incumbent candidate."
Exact(59)
The current Parliament has a slight majority of conservative members.
(For those wondering, a slight majority leaned towards the Republicans).
Sales from foreign carmakers made up a slight majority of the market last year.
Meanwhile, polls show that a slight majority of Americans now favour restrictions on guns.
The vote in the Assembly, where Republicans held a slight majority, was 95-2.
Of these, a slight majority were either noncompliant with guidelines or plainly inappropriate.
By all accounts, the antiwhaling forces retain a slight majority on basic issues.
But Time Warner would have a slight majority on the board of directors and ultimate control.
So will Coke, the slight majority of whose drinks are diet.
The Western-backed coalition, led by Mr. Hariri, holds a slight majority in Parliament.
Polls show that a slight majority of New Jersey voters support same-sex marriage.
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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