Overdue getting the first coffee (5)
I believe the answer is:
latte
'coffee' is the definition.
(type of coffee drink)
'overdue getting the' is the wordplay.
'overdue' becomes 'late' (similar in meaning).
'getting' is an insertion indicator.
'the' becomes 't' (the is pronounced as a 't' sound in some dialects).
'late' going around 't' is 'LATTE'.
'first' acts as a link.
(Other definitions for latte that I've seen before include "Espresso coffee and milk" , "See 21" , "Espresso with frothy milk" , "White coffee" , "Espresso added to frothed steamed milk" .)