The most common and most likely answer to this crossword clue is the 6 letter word OFFALY
O
F
F
A
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Y
More clues leading to the result OFFALY
Buddy explains!

The straight part
We know that "neighbour of Kildare" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case neighbour of kildare = offaly.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.