The most common and most likely answer to this crossword clue is the 3 plus 4 letter word MAD KEEN
I've seen this crossword clue in The Guardian
M
A
D
K
E
E
N
Buddy explains!
The straight part
We know that "Desperately anxious" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case desperately anxious = mad keen.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.