"His pictures were more accurate than photographs, and they had a clarity that made any explanation a waste of words." 'clarity', here, is being used as a noun to describe a characteristic of 'his pictures' ('they'). As a noun, 'clarify' needs an article in English: the ball, a cup, etc. If you substitute other describing-nouns into the sentence you'll see that they need articles to sound correct. e.g. 'a redness', 'a transparency', 'a glow'.

Excellent question. I don't believe the sentence is incorrect without 'a', but it would be unusual to see it as just 'clarity'. This is because the writer is trying to convey the idea that the clarity is an unusually clear kind of clarity. The sentence could be written as 'His pictures were more accurate than photographs, and they had SUCH clarity that IT made any explanation a waste of words' and it would mean more or less exactly the same thing. Clarity is uncountable - you can't have clarities - but there are still different degrees of clarity. For that reason, people use articles when they want to pick one kind of clarity from all the other kinds of clarity. Or, more generally, one version of an uncountable noun from all the other versions. Examples: 'A scarlet that looked more like crimson.' 'A gloom that threatened to envelop us all.'

It's because of the relative clause which follows. You could say simply 'They had clarity.' (in a general sense) without an article. However, the phrase '... that made any explanation a waste of words' is defining the particular type of clarity in question - hence the need for the article.

To answer this question I need to know how the phrase "a clarity" was used in a sentence. Can you provide the full sentence?