The most common and most likely answer to this crossword clue is the 2 plus 8 letter word MY GOODNESS
I've seen this crossword clue in The Guardian
M
Y
G
O
O
D
N
E
S
S
More clues leading to the result MY GOODNESS
Buddy explains!
The straight part
We know that "Well/what might have me beatified" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case well/what might have me beatified = my goodness.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.