'Mere' can be preceded by both definite and indefinite articles, and the same rules of when to use either article apply to the usage of mere. The exception is that the plural of a/an is to use no article at all, but when you use 'mere' with a plural indefinite noun, you use 'a.'
So, since 'three votes' is indefinite (not the three votes), you would say 'a mere three votes.'
"The thought of it makes me sick" becomes "the mere thought of it makes me sick."
"She's a child" becomes "she's a mere child."
"It cost twenty cents" becomes "it cost a mere twenty cents." (twenty cents is indefinite and plural, so it takes the indefinite 'a')
(not a teacher, just a language lover)
Retired English Teacher