Day's beginning sooner than expected, at great cost (6)
I believe the answer is:
dearly
'at great cost' is the definition.
('dearly' can be a synonym of 'at great cost')
'day's beginning sooner than expected' is the wordplay.
'beginning' says to take the initial letters.
'sooner than expected' becomes 'early' ('early' can be a synonym of 'sooner than expected').
The initial letter of 'days' is 'd'.
'd'+'early'='DEARLY'
(Other definitions for dearly that I've seen before include "How I love my wife" , "At a high price - with fondness" , "paying for big mistakes like this!" , "At high cost" , "Extremely - devotedly - at great cost" .)