Black holes are generally the considered as the most symmetric astrophysical bodies in the Universe. If an intermediate mass black hole consumes a nearby red giant and in the process, for a very short time it breaks its highly symmetric geometric property of the horizon. Would such a black hole still have a singularity? Will the singularity be still censored or become naked, breaking the cosmic censorship?
In practice, all black holes observed have veiled any singularities behind a trapped surface. But theoretically, the question has no definitive answer.
If a singularity is to conserve the net angular momentum of all the stuff that has fallen in, then it must not only rotate but have extension in space. Ring and linear singularities are among those which have been studied. A rod-like singularity, spinning like a drum majorette's baton, can in theory penetrate its surrounding trapped surfaces and expose itself as a naked singularity (not sure about rings). Cosmic censorship principles have been postulated but not proven.
The outermost trapped surface is known as the event horizon. It is defined by the gravitational field gradient reaching a critical level at that point. It is easy enough to pull the critical gradient outwards, as for example a binary companion does, but not to push it inwards. The situation is complicated by the fact that the gravitational gradient close to a rotating singularity may differ in different directions. One or more singularities may or may not exist inside it. Again, models differ and some suggest that an actual singularity would take infinite time to form, or at least to appear to form (whatever that can mean when unobservable inside a trapped surface).
So overall, the answer to your question is; yes a distorted event horizon can always sustain a singularity, but there need not be a singularity to be sustained, and even if there is then we cannot rule out its poking outside.
There is a very strong theorem, called the Area increase theorem, which says that in any spacetime where all matter satisfies one of the energy conditions that guarantees that the matter is "normal" (there are several different technical versions of this condition, but think of it as "no negative mass matter"), then it is the case that the total surface area of horizons in that spacdtime will always increase with time.
This means that no matter how much distorting or perturbing you do, you will not be able to rip away an existing horizon from a singularity, and make it naked.
What about Hawking radiation, and black holes that shrink from this, you ask? Well, it turns out that Hawking radiation actually does violate these energy conditions, as a biproduct of its quantum nature.