Make no progress, but hold one's head up mainly (5,5)
I believe the answer is:
tread water
This clue is a double definition.
'make no progress' is the first definition.
'hold one's head up mainly' is the second definition.
(the main is a historical nickname for the sea)
'but' is the link.
(Other definitions for tread water that I've seen before include "Hold back from making progress" , "unwilling to go with the flow?" , "just survive" , "Stay afloat" , "having no solid base" .)