A piece of a mosaic set awkwardly, second time! (7)
I believe the answer is:
tessera
'a piece' is the definition.
(I have seen 'Mosaic piece ' mean 'tessera' so perhaps 'piece' could also mean 'tessera')
'set awkwardly second time' is the wordplay.
'awkwardly' indicates an anagram.
'second' becomes 's' (common abbreviation).
'time' becomes 'era' (era is a kind of time).
'set' anagrammed gives 'tes'.
'tes'+'s'+'era'='TESSERA'
'of a mosaic' is the link.
I am not very happy about this link. Some or all of it may belong to another bit of the clue.
Can you help me to learn more?
(Other definitions for tessera that I've seen before include "Reseats (anag) -- small tile used in a mosaic" , "small bit of artwork" , "only part of the picture" , "Small square tile used in mosaics" , "password" .)