Move stealthily, principally pursuing rare bird (5)
I believe the answer is:
prowl
'move stealthily' is the definition.
(synonyms)
'principally pursuing rare bird' is the wordplay.
'principally' suggests taking the first letters.
'bird' becomes 'owl' (owl is a kind of bird).
The initial letters of 'pursuing rare' is 'pr'.
'pr'+'owl'='PROWL'
(Other definitions for prowl that I've seen before include "Move in a predatory manner" , "Move as predator" , "Walk menacingly" , "Move in a predatory way" , "Steal around predatorily" .)