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This question was inspired by an old post here on meta, about a previous question of the same type that had been removed because it "had too much science fiction", in particular the author added a lot of references to "energy beings" and other (admittely soft! space opera!) "sci-fi" type stuff along with it, although the precise extent as to which is not clear because the original is now gone:

Off topic review: How to feed a small black hole against 1 TW Hawking radiation pressure?

But I thought it was an interesting question in its own right when considered on its own merit, without reference to the "sci-fi" type fluff material. And when I saw it, and note the question is still absent, I wanted to reopen the case, but with the objectionable parts removed or kept to only the minimum necessary to make things clear.

From what I gather of those posts, since the original has been deleted, the question would be: is there some physically possible or engineering-feasible way one could feed a black hole that was of a small enough mass it is emitting 1 TW of Hawking radiation? That is, to get matter in despite the outgoing radiation trying to blow it away?

For the relevant mathematics of this question, we have the black hole lifetime as

$$t_0 = \frac{5120 \pi G^2 M^3}{\hbar c^4}$$

and the radiated power as

$$P = \frac{\hbar c^6}{15360 \pi G^2 M^2}$$

as well as the horizon area:

$$A_H = \frac{16 \pi G^2 M^2}{c^4}$$ For any given power $P$, the relevant mass is seen to be

$$M = \sqrt{\frac{\hbar c^6}{15360 \pi G^2 P}}$$

Taking $P = 10^{12}\ \mathrm{W}$ gives the mass as $1.9 \times 10^{10}$ kg, the lifetime to die completely is 560 Ts (~18 million years), so it at least looks like it won't be vanishing too soon or increasing in power too rapidly. The horizon area is about $10^{-32}\ \mathrm{m^2}$, so the radiation intensity $I$ is $10^{44}\ \mathrm{W/m^2}$ and the radiation pressure (by $p = \frac{I}{c}$) is about $3 \times 10^{35}\ \mathrm{Pa}$. (Note this doesn't take into account near-horizon effects that may change the situation and my GR ain't good enough to work that all out, but you could use this at a few horizon widths for a rough order-of-magnitude estimate. I'd think it will definitely be more than $10^{30}\ \mathrm{Pa}$ of pressure, to really low ball even the order of magnitude, you'll be fighting.).

The question is given the very small throat size and very huge pressure, is there any physically possible way you could force enough matter into it to make its mass grow, and given the "sci-fi" like context of the original question, even just to sustain it as a power source indefinitely radiating 1 TW (which equates to forcing about 11 mg of mass per second into the hole)? This horizon radius is even smaller than an atom. FWIW the concept of a black hole as power source has been suggested by legitimate researchers:

https://en.wikipedia.org/wiki/Black_hole_starship#cite_note-cranewestmoreland2009-2

Or would the black hole have to just be used "as long as it lasts" - that is, after its creation, the only thing that can be done is to sip the energy until it either gets too hot to handle or it finally blows out? (Note that this doesn't necessarily make it useless for running a spacecraft - one could, for example, collect the power to form a beam that would be directed to propel the craft in some form of beamed propulsion perhaps with some kind of Dyson sphere-like construct encasing it, instead of carrying the BH along with it which, would arguably be a better use since the extreme BH mass would not be there to slow/dog it down. Actually if anything a lighter BH might be more suitable for this purpose so you can get much higher output power, but then comes the question of whether it'll die/get too hot before the ship comes up to cruising speed. A 1 PW BH has a lifetime of 18 Gs (about 500 years), which might or might not be enough to get a decent payload up to speed. But then this also makes the feeding problem all the more urgent, and far worse, since the rad pressure goes up rapidly.)

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This started as a comment but it got too big,

Note, that the temperature of the black-hole $$ T_{\mathrm {H} }=\frac {\hbar c^{3}}{8\pi GM} $$ (written in energetic units) with mass $1.9 \times 10^{10}\,\text{kg}$ would be about $0.56\, \text{GeV}$, which would mean that the Hawking radiation would include a lot of pions, muons, protons and corresponding antiparticles.

But this also implies that the equation for the radiated power (the one with $15360^{-1}$) is wrong. It is derived from Stefan–Boltzmann power law, which would be wrong for temperatures at which a lot of massive particles would be radiated. There would be additional channels for radiations of leptons, pions, with each contributing power comparable to EM radiation. So if we define 1 TW as total power of Hawking radiation in channels 'accessible' for the engineering applications (EM radiation, leptons, mesons) and ignoring neutrinos and gravitons that are not readily usable, we would need to correspondingly increase the mass of black hole by some unknown factor. This would lower the temperature and increase horizon area. So from the 'engineering' standpoint the problem becomes somewhat simpler.

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