The most common and most likely answer to this crossword clue is the 6 letter word YAFFLE
I've seen this crossword clue in The Times
Y
A
F
F
L
E
Buddy explains!

The straight part
We know that "Woodpecker" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case woodpecker = yaffle.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 YAFFLE mean?
YAFFLE noun- The European green woodpecker, Picus viridis.
- Of the green woodpecker: to make its distinctive cry.