Follow affair with husband (5)
I believe the answer is:
chase
'follow' is the definition.
(I know that chase is a more specific form of the action follow)
'affair with husband' is the wordplay.
'affair' becomes 'case'.
'with' means one lot of letters goes inside another.
'husband' becomes 'h' (abbreviation).
'case' going around 'h' is 'CHASE'.
(Other definitions for chase that I've seen before include "groove" , "Pursue; decorate (metal) by engraving" , "trail" , "Hound" , "Engrave - hunt - pursue" .)