Anyone know what the problem is?

[mruppal]

Mazdaspeed3

During heavy acceleration on 4th or 5th gear car will have a sputtering sound, like it has troubles accelerating and will jolt back.. If that's the right term to say that the action it does..
Kind of hard to explain, but I hope ^ describes it enough lol. Thanks in advance to anyone with anyone input

[tiger_handheld]

sometimes I cough..a couple of times in a row..at night...

[IMASA]

If you are running a stock tune or a Cobb OTS tune, then the engine is probably knocking so the ECU is pulling timing. For the jolting, your motor/transmission mounts may be shot. Under WOT, the motor does move a bit and will jolt back when you back off the throttle.

Stock tune does not like heavy acceleration in 4th/5th gear at low rpm (under 3k rpm), many folks have bent their rods this way.