The most common and most likely answer to this crossword clue is the 6 plus 9 letter word OLIVER GOLDSMITH
I've seen this crossword clue in The Times
O
L
I
V
E
R
G
O
L
D
S
M
I
T
H
Buddy explains!
The straight part
We know that "Irish write" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case irish write = oliver goldsmith.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.