The most common and most likely answer to this crossword clue is the 5 plus 3 plus 5 letter word SOOTY AND SWEEP
I've seen this crossword clue in The Daily Telegraph
S
O
O
T
Y
A
N
D
S
W
E
E
P
Buddy explains!
The straight part
We know that "Bear with Barker — Corbett had a hand in this popular duo" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case bear with barker — corbett had a hand in this popular duo = sooty and sweep.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.