The most common and most likely answer to this crossword clue is the 6 letter word YAWNER
I've seen this crossword clue in Independent
Y
A
W
N
E
R
Buddy explains!

The straight part
We know that "he might well gape" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case he might well gape = yawner.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 YAWNER mean?
YAWNER noun- A person who yawns.
- (entertainment) Something unexciting or boring that causes yawns, as a book, sporting event, or performance.