The most common and most likely answer to this crossword clue is the 9 letter word PATENTING
I've seen this crossword clue in The Guardian
P
A
T
E
N
T
I
N
G
Buddy explains!
The straight part
We know that "Claiming right to" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case claiming right to = patenting.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 PATENTING mean?
PATENTING verb- To successfully register an invention with a government agency; to secure a letter patent.