Become weary of a complicated rite (4)
I believe the answer is:
tire
'become weary' is the definition.
(get tired)
'complicated rite' is the wordplay.
'complicated' is an anagram indicator (a complex or confusing ordering of the letters).
'rite' is an anagram of 'TIRE'.
'of a' acts as a link.
(Other definitions for tire that I've seen before include "Get weary" , "Weaken" , "Become exhausted" , "Flag" , "Exhaust" .)