Path Correction During GPS Failure

I am trying to program a drone to better understand its flight path when there is a GPS failure. For example, if I tell my drone to fly along a programmed route and along this route there is a curve that is 3/4 of a circle. Let’s say my GPS fails initially at the beginning of the circle. The system I am developing feeds the location to the drone in place of the GPS. However after it travels through this 3/4 circle it realizes that it is in the same place it started. This of course means I’ve made an error. I know the programmed route so I know how far off I am based on the last GPS signal at the beginning of the curve. Is there any research or strategy to feed my pixhawk a false location so that I can correct the path? I cannot communicate with the system in any other way then giving it my current location, so I need to figure out how to get to the correct path by only feeding the system location data.