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A black hole with a Schwarzschild radius of 4 feet would have a mass of 137.5 x Earth and the gravity to go along with it. Such a black hole would instantly liquefy the earth as it shreds it into an accretion disk and shoots gamma rays straight through the earth's poles.

There would be approximately 0 seconds for a committee to consider the problem before being vaporized, thus no solution will be found.

For narrative purposes however, the author might wish to reconsider the mass of the black hole to one that is less instantly catastrophic using the following: calculator. For instance, a black hole with a mass of 72135 metric tons would have a radius of 0.0001071103 femtometers and a lifetime of approximately 1 year before it goes critical and destroys the planet.

Such a black hole would be too small to interact with any atom inside the earth, so it would float freely, ping-ponging back and forth without ever acreting any more mass. A literal ticking-time-bomb if you will as it passes in and out, in and out through the Earth.

Feed the hole negative mass.

Probably you have some negative matter around to power your Alcubierre drive. Load that into your little black hole. You will need a fair bit to make a difference.

To my surprise, this is apparently a done thing in theoretical circles: adding negative mass to model the shrinking of a black hole via Hawking radiation.

https://physics.stackexchange.com/questions/153376/what-would-happen-if-a-negative-mass-crossed-the-event-horizon-of-a-black-hole

The situation of negative mass falling into a black hole does have one
important consequence though, in GR it's the only way for the event
horizon of a black hole to shrink rather than expand, and for this
reason a dynamical black hole metric (the Vaidya metric) with negative
mass falling into it is sometimes used when trying to model the
long-term behavior of a black hole that is "evaporating" due to
continually emitting Hawking radiation (since this is a quantum
effect, and general relativity is not fully compatible with quantum
mechanics, this evaporation should ultimately require a full theory of
quantum gravity to model it completely accurately, but it seems
reasonable to expect that the earlier stages of evaporation, before
the size of the black hole and the energy density approach the Planck
scale where quantum gravity effects are expected to become
significant, should have some close analogue in classical general
relativity).

I will assume you want something vaguely scientifically plausible. (I will also assume when you said "small black hole" you really meant one that is small enough to be contained on the surface of a planet and that the 4' size was not what you really meant.)

To start with, you can't destroy black holes. You can -- with difficulty -- move them. You can feed them and make them more massive. You can charge them and change their spin. But that's about it. So pretty much all you can do is try to get it up out of the planet, and that requires as much thrust as it would to launch the same mass of ordinary material into orbit and a way to apply that thrust to the BH.

(It's true that BHs probably evaporate due to Hawking radiation (though it's never been observed), but if HR is real, evaporation takes a very long time.)

The only way moving the BH might be doable (which wouldn't necessarily destroy the planet as a side-effect) involves charging the BH and then using electromagnetic fields to accelerate it. Pretty hard to do!

That brings up the question of the BH's mass. If it was temporarily contained, then it must have had a large charge and been supported by EM fields. It's mass is its mass, and if it was a small BH -- say the same mass as a mountain -- the containment facility would need to be able to support that mass concentrated into a really, really tiny object. The Schwarzschild radius of a BH of 100,000 tons (which would be very difficult to support with any imaginable technology) would be about 10^-19 meters which is roughly 1 millionth the diameter of a proton.

Something that small would find it very difficult to swallow anything, so while it would orbit inside the planet, it would grow only very, very slowly. It would probably come to a halt (due to dynamical friction) at the planet's core and just sit there growing very, very, very slowly. It would probably have no effect whatsoever on a thousand year timescale.

The bigger the BH, the more of an impact it would have, but the less likely it would be that it could ever have been contained at the planet's surface.

For something that sounds plausible, but totally isn't

Fire charged particles at relativistic speeds on tangents to the event horizon. And just before they cross the event horizon, slow them down to below relativistic speeds with bursts from powerful magnetic and electrical fields.

As they near the black hole, their induced mass creates gravitation fields to compete with the internal mass of the black hole. By firing multiple particles simultaneously at antipodal targets tangential to the event horizon, the black hole can't move so its internal mass it stretched between the equal gravitational forces pulling it apart.

By this means, tiny bits of matter can be teased away from the grip of the black hole.

Keep it up until it is too small to hurt the planet.

Scientists save the world, again. Your welcome.

A black hole with an event horizon circumference of ~13 feet (circumference because diameter is meaningless in a region of space so deeply curved) would mass far more than the Earth. The surface gravity of an Earth-sized planet with such a body at its core would be well beyond survivability of any land animal less tough than a small spider or insect (though aquatic creatures would be okay until they tried to dive too deep -- say, more than about ten meters).

So, don't worry about destroying that four foot black hole. Everyone will be dead before it's known why the Earth is being swallowed -- and it'll never get near the center of the planet; it'll tear the whole planet up before passing without a noticeable change in trajectory.

Spin it up until its angular momentum becomes sufficiently large that it becomes a naked, toroidal singularity, at which point the laws of physics metaphorically throw up their hands and shrug in confusion.