The most common and most likely answer to this crossword clue is the 4 letter word MATT
I've seen this crossword clue in The Sunday Telegraph
M
A
T
T
Buddy explains!

The straight part
We know that "not having brilliant cover" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case not having brilliant cover = matt.The cryptic/wordplay part
In cryptic crossword there's often an wordplay/cryptic part as well.This part involves a more complex hint, such as an anagram, homophone, hidden word, etc.
Sadly, We don't have an explanation for this specific crossword clue yet.
What does MATT mean?
MATT noun- (coppersmithing) An alloy of copper, tin, iron, etc.; white metal.
- Dull; not reflective of light.