The most common and most likely answer to this crossword clue is the 7 letter word QUIBBLE
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More clues leading to the result QUIBBLE
Buddy explains!

The straight part
We know that "little objection" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case little objection = quibble.The cryptic/wordplay part
In cryptic crossword there's often an wordplay/cryptic part as well.This part involves a more complex hint, such as an anagram, homophone, hidden word, etc.
Sadly, We don't have an explanation for this specific crossword clue yet.
What does QUIBBLE mean?
QUIBBLE noun- A pun.
- An objection or argument based on an ambiguity of wording or similar trivial circumstance; a minor complaint.
- To complain or argue in a trivial or petty manner.