= Solution
A small <loss cone> does not by itself specify its supply rate. Introduce a characteristic orbital radius $r$, an <orbital period> $P\sim r/v$, the number $N_r$ of stars on the relevant energy range, and their local number density $n$. Weak encounters of <impact parameter> $b$ give velocity kicks of order $Gm/(bv)$. Summing their squared kicks over encounter rate $2\pi b\,db\,nv$ gives
$$
\frac{d\langle\Delta v^2\rangle}{dt}
\sim\frac{G^2m^2n\log\Lambda}{v},
\qquad
\boxed{t_{\rm rel}\sim\frac{v^3}{G^2m^2n\log\Lambda}.}
$$
Here $\log\Lambda$ is the <Coulomb logarithm in stellar dynamics>. This derives the <stellar relaxation time> relevant to angular-momentum scattering, up to the velocity-averaging constants.
Let $J_c^2\sim GM_Hr$ and define the dimensionless cone size $\mathcal R_{\rm lc}=J_{\rm lc}^2/J_c^2\sim R_{\rm sch}/r$. In one period, <two-body relaxation> changes $J^2$ by order $J_c^2P/t_{\rm rel}$, so the refilling parameter is
$$
q\sim\frac{P}{t_{\rm rel}\mathcal R_{\rm lc}}.
$$
If refilling is sufficiently rapid to keep the cone populated, an isotropic fraction of order $\mathcal R_{\rm lc}$ reaches its absorbing pericentre each orbit. The full-cone rate is
$$
\boxed{\Gamma_{\rm full}\equiv-\dot N_r
\sim\frac{N_r\mathcal R_{\rm lc}}P
\sim\frac{N_r}P\frac{R_{\rm sch}}r.}
$$
If $q\ll1$, the cone is depleted and capture is diffusion-limited. At fixed orbital energy, the two transverse angular-momentum coordinates have an approximately two-dimensional diffusion equation. Its steady radial solution between the absorbing boundary $J_{\rm lc}$ and the reservoir $J_c$ satisfies
$$
\frac1J\frac d{dJ}\left(J\frac{df}{dJ}\right)=0,
\qquad f(J)\propto\frac{\log(J/J_{\rm lc})}{\log(J_c/J_{\rm lc})}.
$$
The gradient supplies a constant inward flux. With diffusion coefficient of order $J_c^2/t_{\rm rel}$ and reservoir population $N_r$, this gives
$$
\boxed{\Gamma_{\rm empty}\sim
\frac{N_r}{t_{\rm rel}\log(1/\mathcal R_{\rm lc})}.}
$$
Constants of two depend on using $J$ or $J^2$ and on the definition of $t_{\rm rel}$. A rough interpolation uses the slower of the orbital draining and diffusion-supply rates; an accurate transition requires an orbit-averaged diffusion calculation. These are the <empty and full loss-cone capture rates>. In particular, multiplying $N_r/t_{\rm rel}$ by the cone's tiny solid-angle fraction would incorrectly treat diffusive feeding as independent isotropic redraws.
For a one-zone black-hole-dominated cluster with $N_r\sim N$, $n\sim N/r^3$ and $v^2\sim GM_H/r$,
$$
t_{\rm rel}\sim\frac{(M_H/m)^2}{N\log\Lambda}P,
\qquad
\Gamma_{\rm empty}\sim\frac{N^2\log\Lambda}{(M_H/m)^2P\log(r/R_{\rm sch})}.
$$
The mass ingestion rate is $\dot M_H\simeq m\Gamma$. A radially extended cluster requires summing the energy-dependent fluxes, not replacing every radius by the same $r$. \b[No numerical capture rate follows from $N,m,M_H$ alone:] its radius/density, orbital distribution and refilling regime are required. The displayed relations are the requested estimates with those physical scales explicit.
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