Petty quarrel about rhyme's opening character -- Jack, perhaps (5)
I believe the answer is:
sprat
'jack perhaps' is the definition.
(nursery rhyme character Jack Sprat)
'petty quarrel about rhyme's opening character' is the wordplay.
'petty quarrel' becomes 'spat' (a spat is a disagreement or argument).
'about' means one lot of letters goes inside another.
'opening character' indicates taking the first letters.
The first letter of 'rhymes' is 'r'.
'spat' placed around 'r' is 'SPRAT'.
(Other definitions for sprat that I've seen before include "'Set a . . . . . to catch a herring'" , "Couple sharing meat" , "Small fish (used to catch mackerel?)" , "Small herring-type fish" , "Herring's smaller relative" .)