The most common and most likely answer to this crossword clue is the 6 letter word CHESTY
C
H
E
S
T
Y
Buddy explains!

The straight part
We know that "having a wheeze" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case having a wheeze = chesty.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 CHESTY mean?
CHESTY adjective- (of a woman) Having large breasts; busty.
- (of a cough or cold) Not dry; involving the coughing of phlegm.
- Coming from, or associated with, the chest.