Lives with girl? That’s the problem! (5)
I believe the answer is:
issue
'the problem' is the definition.
(thesaurus)
'lives with girl?' is the wordplay.
'lives' becomes 'is' ('be' can be a synonym of 'live').
'with' says to put letters next to each other.
'girl?' becomes 'sue' (**).
'is'+'sue'='ISSUE'
'that's' is the link.
(Other definitions for issue that I've seen before include "Put out; topic for discussion" , "Progeny of, send out" , "particular edition of magazine?" , "Major topic for debate" , "Topic; give out" .)