  We evaluate the validity of the weak form of the cosmic censorship conjecture
and the third law of black hole dynamics for Kerr-MOG black holes interacting
with scalar test fields. Ignoring backreaction effects, we first show that both
extremal and nearly extremal Kerr-MOG black holes can be overspun into naked
singularities by scalar test fields with a frequency slightly above the
superradiance limit. In addition, nearly extremal Kerr-Mog black holes can be
continuously driven to extremality by test fields. Next, we employ backreaction
effects based on the argument that the angular velocity of the event horizon
increases before the absorption of the test field. Incorporating the
backreaction effects, we derive that the weak form of the cosmic censorship and
the third law are both valid for Kerr-Mog black holes with a modification
parameter $\alpha \lesssim 0.03$, which includes the Kerr case with $\alpha=0$.
