The most common and most likely answer to this crossword clue is the 8 letter word INSULATE
I've seen this crossword clue in The Daily Telegraph
I
N
S
U
L
A
T
E
More clues leading to the result INSULATE
- Lag having arrived behind schedule holds us back
- Lag back at 1am, after consuming drink endlessly
- Cover up offensive remark about a European
- Lag‘s offence receiving answer by end of sentence
- Latin island’s empty theatre cut off from communication
- Protect from rudeness about American by English
- One in slight trouble, finally gets cover
- Cushion Brown left in Oman, possibly
Buddy explains!

The straight part
We know that "Cover up" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case cover up = insulate.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 INSULATE mean?
INSULATE verb- To separate, detach, or isolate.
- To separate a body or material from others, e.g. by non-conductors to prevent the transfer of electricity, heat, etc.