Trouble over one offspring’s secret affair (7)
I believe the answer is:
liaison
'secret affair' is the definition.
(I've seen this before)
'trouble over one offspring's' is the wordplay.
'trouble' becomes 'ail' ('ail' can be a synonym of 'trouble').
'over' shows that the letters should be reversed in order.
'one' becomes 'i' (Roman numeral).
'offspring' becomes 'son' (son is a kind of offspring**).
'ail' in reverse letter order is 'lia'.
'lia'+'i'+'son'='LIAISON'
(Other definitions for liaison that I've seen before include "Co-operation; love affair" , "Communication between people or organisations" , "Intimate relationship" , "entanglement" , "Collaboration" .)