The most common and most likely answer to this crossword clue is the 6 letter word THATCH
I've seen this crossword clue in The Times
T
H
A
T
C
H
More clues leading to the result THATCH
- Protective cover Queen got from prime minister
- Prime minister cutting Her Majesty’s hair
- PM ignoring the Queen? That could be hairy!
- Roofing material in yonder church
- First of two children could give you a shock
- Barnet — specifically the church there
- Roofing material which covers church
- Feature of Trump and PM putting the Queen off
- Potentially unruly hair of PM? No hesitation
- Giant hat chosen to accommodate unruly head of hair
Buddy explains!

The straight part
We know that "Barnet" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case barnet = thatch.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 THATCH mean?
THATCH noun- Straw, rushes, or similar, used for making or covering the roofs of buildings, or of stacks of hay or grain.
- Any of several kinds of palm, the leaves of which are used for thatching.
- A buildup of cut grass, stolons or other material on the soil in a lawn.
- (by extension) Any straw-like material, such as a person's hair.
- To cover the roof with straw, reed, leaves, etc.