Precious little time — almost seven days! (4)
I believe the answer is:
twee
'precious' is the definition.
(I've seen this before)
'little time almost seven days' is the wordplay.
'little time' becomes 't' (abbreviation for time).
'almost' means to remove the last letter (almost the whole word is kept).
'seven days' becomes 'week' (I've seen this before).
'week' with its last letter removed is 'wee'.
't'+'wee'='TWEE'
(Other definitions for twee that I've seen before include "Excessively sweet" , "rather quaint" , "Precious - sentimental" , "Excessively cute" , "Affectedly dainty or refined" .)