The most common and most likely answer to this crossword clue is the 4 letter word LIEF
I've seen this crossword clue in The Guardian
L
I
E
F
Buddy explains!

The straight part
We know that "gladly" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case gladly = lief.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 LIEF mean?
LIEF adjective- Beloved, dear, agreeable.
- Ready, willing.
- Readily, willingly, rather.
- The fibre by which the petioles of the date palm are bound together, from which various kinds of cordage are made.