If you're thinking that the matter and antimatter have gravity and that gravity disappears when they annihilate because their mass is gone, that's not the case. All energy and momentum produces gravity, not just mass.
You could only modulate gravity as fast as you could move energy close to you or far away from you. An implausibly powerful gamma ray flashlight flicking on and off would let you do this. Get the beam powerful enough, and it will have a gravitational field.
Get the beam powerful enough and it will have meaurable gravitational field. All beams of any intensity technically do contribute some, albeit infinitesimal, component to the gravitational field.
It's a fun exercise to calculate just how strong a laser beam you'd need to have to make a black hole entirely out of photons.
A black hole can actually cause an additional singularity due to the blueshifting of infalling light. IIRC, one researcher studied this phenomenon and called it a "blue sheet."
If you're thinking that the matter and antimatter have gravity and that gravity disappears when they annihilate because their mass is gone, that's not the case. All energy and momentum produces gravity, not just mass.