The most common and most likely answer to this crossword clue is the 3 letter word LID
I've seen this crossword clue in The Times
L
I
D
More clues leading to the result LID
Buddy explains!

The straight part
We know that "cover" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case cover = lid.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 LID mean?
LID noun- A thin skin membrane that covers and moves over an eye.
- The top or cover of a container.
- A cap or hat.
- One ounce of cannabis.
- A bodyboard or bodyboarder.
- A motorcyclist's crash helmet.
- In amateur radio, an incompetent operator.
- A hermetically sealed top piece on a microchip such as the integrated heat spreader on a CPU.
- A restraint or control, as when "putting a lid" on something.
- To put a lid on (something).