The cables go into a grooved wheel on the right side of the carb under that big silver cover on that bike. They each have an inline adjuster for setting the slack (about 1/8" rotation before engaging the throttle wheel). And yeah, looks like you need a new cable if that white piece is broken. Try to find a Motion Pro cable. Cheaper than OEM but great quality.
Just for GP, when you take it apart, post your specifics on this bike-how many hour ridden-and if either or both of the cable ends were frayed at the carb end.