The most common and most likely answer to this crossword clue is the 7 letter word SPILLED
I've seen this crossword clue in The Daily Telegraph
S
P
I
L
L
E
D
Buddy explains!
The straight part
We know that "admitted the truth, say" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case admitted the truth, say = spilled.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 SPILLED mean?
SPILLED verb- To drop something so that it spreads out or makes a mess; to accidentally pour.
- To spread out or fall out, as above.
- To drop something that was intended to be caught.
- To mar; to damage; to destroy by misuse; to waste.
- To be destroyed, ruined, or wasted; to come to ruin; to perish; to waste.
- To cause to flow out and be lost or wasted; to shed.
- To cover or decorate with slender pieces of wood, metal, ivory, etc.; to inlay.
- To relieve a sail from the pressure of the wind, so that it can be more easily reefed or furled, or to lessen the strain.
- (Australian politics) To open the leadership of a parliamentary party for re-election.
- To reveal information to an uninformed party.
- (of a knot). To come undone.