The most common and most likely answer to this crossword clue is the 11 letter word SCHOLARSHIP
I've seen this crossword clue in The Times
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Buddy explains!

The straight part
We know that "Award given to student/exceptional erudition" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case award given to student/exceptional erudition = scholarship.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 SCHOLARSHIP mean?
SCHOLARSHIP noun- A grant-in-aid to a student.
- The character or qualities of a scholar.
- The activity, methods or attainments of a scholar.
- The sum of knowledge accrued by scholars; the realm of refined learning.
- The first year of high school, often accompanied by exams that needed to be passed before advancement to the higher grades.
- To attend an institution on a scholarship.
- To grant a scholarship to.