Voltage drops on circuit powering a servo motor. Is there a solution?
You basically had the right idea to use a cap for voltage stabilisation. But a cap with 4F capacity is a supercapacitor, and those have a very high output resistance. So your Supercap is not able to deliver the current needed for the servo. 

Please try to use a few 220uF aluminium electrolytic type caps. They have output resistance in the milliohms range and they should do the trick of stabilizing your voltage.