The most common and most likely answer to this crossword clue is the 6 letter word GAMELY
I've seen this crossword clue in The Times
G
A
M
E
L
Y
More clues leading to the result GAMELY
Buddy explains!

The straight part
We know that "making valiant effort" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case making valiant effort = gamely.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 GAMELY mean?
GAMELY adjective- Sportive; lively; joyful.
- (hunting, fishing, etc.) Of or pertaining to game.
- In a game or plucky manner; in a willing and spirited fashion.