Is this sentence correct?
"He was a badly brought up child."
Normally an article comes with adjective.
But badly is an adverb. Can we use article before it?
You must, in fact, put it there. "Badly brought up" is a phrase that uses a verb form (sometimes called a verbal) as an adjective. We must use an adverb because "brought" is a verb, but the function of the whole phrase is as an adjective, so it's no different grammatically than saying "a sad child".
To put "a", for example, in front of "brought up" here would change the meaning, if it meant anything at all. The whole adjectival phrase "badly brought up" is indivisible.