Thing that’s crossed channel that is crossing river? Correct (4,4)
I believe the answer is:
idee fixe
'thing that's crossed channel' is the definition.
(French term for an obsession or 'thing')
'that is crossing river? correct' is the wordplay.
'that is' becomes 'i.e.' (short for Latin 'id est', 'that is').
'crossing' is an insertion indicator.
'river?' becomes 'dee' (River Dee).
'correct' becomes 'fix' (fix can mean to improve or correct).
'dee'+'fix'='deefix'
'ie' placed around 'deefix' is 'IDEE FIXE'.
(Other definitions for idee fixe that I've seen before include "Obsession" , "Nothing else comes to mind" .)