Catch first woman not quite backing people on a strict diet (6)
I believe the answer is:
vegans
'people on a strict diet' is the definition.
(vegans don't eat animal products)
'catch first woman not quite backing' is the wordplay.
'catch' becomes 'snag' (snagging is a kind of catching).
'first woman' becomes 'Eve' (first woman in the bible).
'not quite' means to remove the last letter.
'backing' is a reversal indicator.
'eve' with its last letter removed is 'ev'.
'snag'+'ev'='snagev'
'snagev' backwards is 'VEGANS'.
(Other definitions for vegans that I've seen before include "They eat no animal products" , "People who don't want certain food" , "Very strict vegetarians" , "Extreme vegetarians" .)