Does an uncountable noun take an article if it has an adjective before it?

We all know that uncountable noun does not take an article. But then, if there's an adjective modifying an uncountable noun what should happen?

An article before adjective + uncountable noun? OR
No article before adjective + uncountable noun?

My research:

I found some examples of having article + adjective + uncountable noun such as...

A soothing music.
A deadly pollution.
A hot tea.
A hard work. And the like.

Answer

I already chose oerkelens' answer but I found something useful while reading Swan's Book. I think it's very helpful and thus answering here.

With certain uncountable nouns – especially nouns referring to human emotions and mental activity – we often use a/an when we are limiting their meaning in some way (and we generally limit it by putting adjectives). Here are the examples -

We need a secretary with a first-class knowledge of German (NOT... with first class knowledge of German).
She has always had a deep distrust of strangers.
My parents wanted me to have a good education (NOT .... to have good education).

(Adapted from Practical English Usage, section 149.4)