On her orders, shoot down the Mutant Primate, avoiding the mercenaries it knocks back at you. When it has lost a 1/4 of its health, it will throw the ape guards back at you. Move out of the way (when you see a tiny red circle) to avoid the apes. When it's health gets halfway down, jump over to a small inlet.
The primate sends in Alligator-Crab-Kangaroo Mutant to deter your shooting. They will occasionally drop health pickups that you can use when necessary. Take them out quickly, then continue to shoot the primate. Once it's health gets 3/4 of the way down, hop off to a cliff where the beast will confront you directly.
It's going to try to smash the ground and claw you. Mega-Jump to avoid both of them. Once you get the primate's health down to zero, it'll let go of Sly. If you die at any point in the job, the Primate will drop Sly into its mouth & he will die.