The most common and most likely answer to this crossword clue is the 4 plus 3 letter word TINY TIM
I've seen this crossword clue in The Guardian
T
I
N
Y
T
I
M
Buddy explains!
The straight part
We know that "He made an unexceptional plea for blessing" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case he made an unexceptional plea for blessing = tiny tim.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.