If the emission has not flattened, then the Poisson solver would only get down to some, perhaps not low enough, level.
If the emitted current has not settled, try the following:
Reduce the cycle-to-cycle averaging factor. The averaging factor often needs to be less than 0.5 for higher density meshes using the quadratic field solver.
If field emission, and the setting above does not flatten the emission, then introduce a cycle-dependent function to the emitted current fraction.
If the Poisson solver is not settled, try the following:
Add more cycles if the trend is decreasing.
Reduce the Poisson Solver Tolerance.
Increase the number of particles per emission location.
Improve the mesh in the emission region.
Consider a quadratic field solution, with perhaps an even lower Poisson Solver Tolerance.