Become greater with a sincere shift (8)
I believe the answer is:
increase
'become greater' is the definition.
(I've seen this before)
'with a sincere shift' is the wordplay.
'with' means one lot of letters goes inside another.
'shift' indicates anagramming the letters.
'sincere' with letters rearranged gives 'increse'.
'a' inserted into 'increse' is 'INCREASE'.
(Other definitions for increase that I've seen before include "boost" , "Grow in size" , "Grow bigger" , "Become bigger or greater in amount" , "Multiply" .)