The most common and most likely answer to this crossword clue is the 3 plus 5 letter word LAY CLAIM
L
A
Y
C
L
A
I
M
Buddy explains!
The straight part
We know that "make an assertion" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case make an assertion = lay claim.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.