After first of January, one’s having to live in digs (5)
I believe the answer is:
jibes
'digs' is the definition.
('jibe' can be a synonym of 'dig')
'after first of january one's having to live' is the wordplay.
'after' says to put letters next to each other.
'first of january' becomes 'j' (the first letter of the word 'January').
'one's' becomes 'is' (roman numeral).
'having' indicates putting letters inside.
'to live' becomes 'be' ('be' can be a synonym of 'live').
'is' enclosing 'be' is 'ibes'.
'j' next to 'ibes' is 'JIBES'.
'in' is the link.
(Another definition for jibes that I've seen is " Insulting or mocking remarks".)