Continue abandoning area and cross river (4)
I believe the answer is:
styx
'river' is the definition.
(mythical river)
'continue abandoning area and cross' is the wordplay.
'continue' becomes 'stay' (staying is a kind of continuing).
'abandoning' suggests deleting specific letters.
'area' becomes 'a' (maths abbreviation).
'and' is a charade indicator (letters next to each other).
'cross' becomes 'x' (X is cross-shaped).
'stay' with 'a' taken away is 'sty'.
'sty'+'x'='STYX'
(Other definitions for styx that I've seen before include "Underworld river ferried by Charon" , "River separating Earth from Hades" , "River crossed at death" , "The last river one crosses" , "Underground river" .)