The number that you submitted (8566934B) is the part number for the owner's manual of several different models of Kenmore washers. From your description, I take it that you have a fabric softener dispenser as shown in the first image below. If this is the type of fabric softener dispenser that you have on your washer, make sure that you thoroughly clean the dispenser using the instructions in the image below. This will allow the dispenser to drain properly.
Follow the instructions in your owner's manual regarding the dilution of liquid fabric softener as well. Most liquid fabric softeners are so thick that they will not dispense properly unless they are diluted with warm water. Pour some fabric softener into the dispenser and then add warm water to fill the dispenser to a level just below the lowest edge in the dispenser opening. If you were using just full strength fabric softener in the dispenser, then diluting it should help it dispenser properly.
When the washer spins for the first time (prior to the rinse cycle), the fabric softener solutions is slung up and out of the dispenser cone-shaped cup and is held under the lip of the dispenser top by the centrifugal force of the spinning agitator. When the spinning stops so that the washer can fill for the rinse cycle, the softener solution drops down the outside of the dispenser and into the washer tub. Understanding this process may help you determine the cause of your problem. Thoroughly cleaning the dispenser and diluting the fabric softener should .
If you need more help, resubmit your question with additional details including the full model number of your Kenmore washer.