if I drop a mass into a black hole, will the black hole's gravity be asymmetrical before the mass reaches the singularity?

As the falling object approaches the horizon from outside, the situation is already not spherically symmetric so neither is the spacetime. In a reference frame in which the object passes the horizon in a finite time, the spacetime outside remains non-spherically symmetric after the object passes the horizon, though the asymmetry will be dynamic and will tend to radiate away until eventually the whole thing settles to a symmetric shape (in the absence of angular momentum). At all times the spacetime at any given place is influenced only by matter and signals which can arrive there, in the future light cone of other events. So it is not influenced by events beyond the horizon.