Affair involving Yankee swimmer's sport (3-7)
I believe the answer is:
Here is my best explanation:
'sport' is the definition.
(fly-fishing is a kind of sport)
'affair involving yankee swimmer's' is the wordplay.
'affair' becomes 'fling' (I have seen 'Brief affair' mean 'fling' so perhaps 'affair' could also mean 'fling').
'involving' means one lot of letters goes inside another.
'yankee' becomes 'y' (phonetic alphabet: alpha, bravo, charlie etc.).
'swimmer' becomes 'fish' (fish is a kind of aquatic vertebrate**).
'fling' placed around 'yfish' is 'FLY-FISHING'.