The most common and most likely answer to this crossword clue is the 5 plus 4 letter word MODAL VERB
I've seen this crossword clue in Independent
M
O
D
A
L
V
E
R
B
Buddy explains!
The straight part
We know that "(grammatically, again,) ‘may’, ‘could’ and ‘should’ have provided examples (of MODAL VERBS)" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case (grammatically, again,) ‘may’, ‘could’ and ‘should’ have provided examples (of modal verbs) = modal verb.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.