Offspring lives with us, rising expenses primarily (5)
I believe the answer is:
issue
'offspring' is the definition.
(I know that offspring can be written as issue)
'lives with us rising expenses primarily' is the wordplay.
'lives' becomes 'is' ('be' can be a synonym of 'live').
'with' is a charade indicator (letters next to each other).
'rising' shows that the letters should be reversed in order (going up in a down clue).
'primarily' says to take the initial letters.
The initial letter of 'expenses' is 'e'.
'us' back-to-front is 'su'.
'is'+'su'+'e'='ISSUE'
(Other definitions for issue that I've seen before include "Problem - child" , "Topic; give out" , "The children to come forth" , "Regular publication" , "particular edition of magazine?" .)