The most common and most likely answer to this crossword clue is the 4 letter word RAMI
R
A
M
I
Buddy explains!

The straight part
We know that "Weavers’ fibre" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case weavers’ fibre = rami.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 RAMI mean?
RAMI noun- A small spray or twig.
- A branching, as of nerves or blood vessels.
- The stem of a barb of a feather, from which the barbules extend.
- A bony projection, particularly of the jaw, but also in the groin area, both subject to the maturing process of symphysis.