The most common and most likely answer to this crossword clue is the 7 letter word LIONIZE
I've seen this crossword clue in Independent
L
I
O
N
I
Z
E
Buddy explains!

The straight part
We know that "Glorify" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case glorify = lionize.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 LIONIZE mean?
LIONIZE verb- To treat (a person) as if they were important, or a celebrity.
- To visit famous places in order to revere them.
- To behave as a lion.