The Battle of Barrett's Farm did not take place during the Vicksburg Campaign. It took place on July 6, 1864, when Union General Slocum was attacked by Confederate forces commanded by General Wirt Adams. Slocum was returning to Vicksburg after an expedition to Jackson, and was attacked on the evening of the 6th on the outskirts of Clinton just east of the town. It was a bloody little fight given the number of participants involved and the short time the fighting lasted. There is a brief description of the fight here: http://www.mississippiscv.org/MS_Units/1st_mississippi_light_artillery.htm. As I remember the Vicksburg Daily Herald has several articles in July 1864 about the fighting that took place during Slocum's expedition to Jackson, and I think there is an article specifically about the fighting at Barrett's Farm.
I hope this helps, and if you have any questions, please just let me know.

Jeff