Obviously, there is some irregularity in the motion of the movement's gear train. Possibly the problem is in the escapement. Ordinarily each swing or "beat" of a mechanical timepiece's balance wheel will only allow a single escape wheel tooth to "escape" from between the pallet lever jewels or pins so that the gear train can advance a bit. However, excessive wear on the teeth of the escape wheel and on steel pallet pins can occasionally allow an extra tooth to escape and produce a "jerky" motion of the second hand. This problem is far more common in watches that have metal pallet pins instead of jewels and in which the thin coating of lubricant that protects the contacting metal surfaces from friction is no longer working.
IF this is the cause of your watches' problems, then you will have to replace the escape wheels and pallet forks. It's really easier to just scrap the entire movment and replace it with a less worn one that, hopefully, IS properly lubricated.