The most common and most likely answer to this crossword clue is the 7 plus 8 letter word CHASING RAINBOWS
I've seen this crossword clue in The Guardian
C
H
A
S
I
N
G
R
A
I
N
B
O
W
S
Buddy explains!
The straight part
We know that "Delusional activity," is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case delusional activity, = chasing rainbows.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.