The most common and most likely answer to this crossword clue is the 6 plus 4 plus 2 letter word LAVISH CARE ON
I've seen this crossword clue in The Guardian
L
A
V
I
S
H
C
A
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O
N
Buddy explains!
The straight part
We know that "Go out of one’s way to look after" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case go out of one’s way to look after = lavish care on.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.