Bird: that thing below moving target (5,3)
I believe the answer is:
great tit
'bird' is the definition.
(I've seen this before)
'thing below moving target' is the wordplay.
'thing' becomes 'it'.
'below' means one lot of letters go next to another (in a down clue, some letters go below others).
'moving' indicates an anagram (letters are moved around).
'target' is an anagram of 'greatt'.
'it' after 'greatt' is 'GREAT TIT'.
'that' is the link.
(Other definitions for great tit that I've seen before include "Small songbird" , "Common small bird with yellow breast" , "A bird; target it (anag)" , "Black-headed bird" , "Bird changed" .)