Moment that's doubtful on the first of January (5)
I believe the answer is:
jiffy
'moment' is the definition.
(I know that jiffy is a type of moment)
'doubtful on the first of january' is the wordplay.
'doubtful' becomes 'iffy' (I've seen this before).
'on' is a charade indicator (letters next to each other).
'the first of january' becomes 'J' (1st letter of 'january').
'iffy' after 'j' is 'JIFFY'.
'that's' is the link.
(Other definitions for jiffy that I've seen before include "Moment (slang)" , "Split second" , "An instant" , "Very short moment - type of bag" , "Two shakes of a lamb's tail?" .)