If you start your round adjacent to an opponent, can you use the Leaping Attack ability (of the Mobile Fighter variant) after taking a 5-foot step and making a full-attack against that opponent?
Is the attack and damage bonus gained only for the first attack, or for all the attacks (of the full-attack action)?
You cannot move in any round that you use a 5-foot step, nor can you use a 5-foot step in any round in which you've moved.
Jumping is generally part of movement (it can even be part of a 5-ft. step I suppose).
As such, I'm guessing the answer to your question is "no."
EDIT: Just read Leaping Attack. Despite its name, it has nothing to do with jumping. Yes, a 5-ft. step would indeed qualify you for the bonuses. My mistake.
You would gain the bonuses on ALL your attack and damage rolls (I assume for the round, if not until your next turn).