Nearly everybody backs the favourite - that's a bit of a bloomer
I believe the answer is:
petal
'a bit of a bloomer' is the definition.
(I've seen this before)
'nearly everybody backs the favourite' is the wordplay.
'nearly' means to remove the last letter.
'everybody' becomes 'all' (all people).
'backs' says to put letters next to each other.
'the favourite' becomes 'pet' (synonyms).
'all' with its final letter taken off is 'al'.
'al' after 'pet' is 'PETAL'.
'that's' acts as a link.
(Other definitions for petal that I've seen before include "Flower part seen on plate" , "Corolla leaf" , "Floral segment" , "Leapt (anag.)" , "Leaf-like part of flower head" .)