The most common and most likely answer to this crossword clue is the 9 letter word DANDIPRAT
I've seen this crossword clue in The Guardian
D
A
N
D
I
P
R
A
T
More clues leading to the result DANDIPRAT
Buddy explains!

The straight part
We know that "Old three-halfpenny" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case old three-halfpenny = dandiprat.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 DANDIPRAT mean?
DANDIPRAT noun- An English coin worth three half-pence
- An insignificant or contemptible person