The most common and most likely answer to this crossword clue is the 2 plus 3 plus 4 letter word ON THE MAKE
I've seen this crossword clue in Independent
O
N
T
H
E
M
A
K
E
Buddy explains!
The straight part
We know that "Eager for promotion" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case eager for promotion = on the make.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.