If I play a Temporal Breach on a unit that my opponent played from my discard with Kharox and is now on a battlefield, will my unit be played on that battlefield?

Yes — you get the unit back and it's played to that battlefield.

When Temporal Breach resolves, it banishes the unit and then instructs its owner (you) to play it back to the same location. That location is a battlefield, and you're allowed to play a unit there even though your opponent controls it — doing so applies Contested status (Rules 190.3.a, 355.2). Ownership never changed when Kharox played your unit from your trash; Kharox's controller only gained control of it on the board.

If your opponent still has units at that battlefield, a Combat Showdown stages against them once play resumes. If they've moved their other units away, it's just a Non-Combat Showdown. The one case where this breaks is if the unit was in your opponent's Base — you can't legally play a unit to an opponent's base, so that instruction is ignored and the unit stays banished (FAQ 12422; Rules 359.3.e.6, 355.2.a).