The most common and most likely answer to this crossword clue is the 4 letter word WRAP
I've seen this crossword clue in Independent
W
R
A
P
More clues leading to the result WRAP
- Responsibility taken after wife stole
- Stole hit repeatedly heard on radio?
- Hit, say, to cover
- Gift paper maybe will revolutionise any present at the outset
- Stole whiskey, facing charge
- Cover charge for audience
- What to do to present western music style
- Completion of filming western followed by strike
- Cover sound like a 50 cent song
- Cover hand up, bandaging end of finger
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 = wrap.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 WRAP mean?
WRAP verb- To enclose (an object) completely in any flexible, thin material such as fabric or paper.
- To enclose or coil around an object or organism, as a form of grasping.
- To conceal by enveloping or enfolding; to hide.
- (video production) To finish shooting (filming) a video, television show, or movie.
- (lines, words, text, etc.) To break a continuous line (of text) onto the next line
- To make functionality available through a software wrapper.
- To (cause to) reset to an original value after passing a maximum.
- A garment that one wraps around the body to keep oneself warm.
- A type of food consisting of various ingredients wrapped in a tortilla or pancake.
- (entertainment) The completion of all or a major part of a performance.
- A wraparound mortgage.
- A sharp blow with something hard.
- Blame for something.
- A charge, whether or not it results in a conviction.
- A casual talk.
- Rap music.
- A song, verse, or instance of singing in the style of rap music.
- An appraisal.
- A positive appraisal; a recommendation.
- A lay or skein containing 120 yards of yarn.
- Any of the tokens that passed current for a halfpenny in Ireland in the early part of the eighteenth century; any coin of trifling value.
- A whit; a jot.