Family stopping small cat doing unhealthy activity (7)
I believe the answer is:
smoking
'activity' is the definition.
(smoking is a kind of activity)
'family stopping small cat' is the wordplay.
'family' becomes 'kin' (synonyms).
'stopping' indicates putting letters inside (stopping can mean blocking a gap).
'small' becomes 's' (abbreviation - e.g. clothes size).
'cat' becomes 'mog' (informal British term).
's'+'mog'='smog'
'kin' going into 'smog' is 'SMOKING'.
'doing unhealthy' is the link.
This may not be right. Some or all of it may be part of another bit of the clue.
Can you help me to learn more?
(Other definitions for smoking that I've seen before include "Puffing a fag" , "Using tobacco" , "unhealthy habit" , "Emitting a cloud of gas" , "Having a cigarette" .)