The most common and most likely answer to this crossword clue is the 7 letter word WORKOUT
I've seen this crossword clue in The Guardian
W
O
R
K
O
U
T
More clues leading to the result WORKOUT
Buddy explains!

The straight part
We know that "training session" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case training session = workout.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 WORKOUT mean?
WORKOUT noun- An exercise session; a period of physical exercise.
- A schedule or program of specific exercises, especially one intended to achieve a particular goal.
- (by extension) Any activity that requires much physical or mental effort, or produces strain.