Gone to pick up tea inside (or coffee) (5)
I believe the answer is:
latte
'coffee' is the definition.
(I know that latte is a type of espresso)
'gone to pick up tea' is the wordplay.
'gone' becomes 'late' (both can mean dead).
'to pick up' means one lot of letters goes inside another.
'tea' becomes 't' (I've seen this in other clues).
'late' placed around 't' is 'LATTE'.
'inside or' acts as a link.
I am not very happy about this link. Some or all of it may belong to another bit of the clue.
(Other definitions for latte that I've seen before include "Milky coffee drink" , "Italian word for milk" , "Style of coffee" , "Espresso with frothy milk" , "Coffee beverage" .)