The most common and most likely answer to this crossword clue is the 9 letter word UNDERTOOK
I've seen this crossword clue in The Guardian
U
N
D
E
R
T
O
O
K
Buddy explains!
The straight part
We know that "Promised to deal with" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case promised to deal with = undertook.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 UNDERTOOK mean?
UNDERTOOK verb- To take upon oneself; to start, to embark on (a specific task etc.).
- To commit oneself (to an obligation, activity etc.).
- To overtake on the wrong side.
- To pledge; to assert, assure; to dare say.
- To take by trickery; to trap, to seize upon.
- To assume, as a character; to take on.
- To engage with; to attack, take on in a fight.
- To have knowledge of; to hear.
- To have or take charge of.