= Solution
Only the <photon> energy above the <hydrogen> <ionization> threshold initially becomes <photoelectron> <kinetic energy>. The stated mean therefore supplies
$$
E_{\rm heat}=\langle h\nu\rangle-h\nu_{\rm Lye}=k_BT_*.
$$
Let $\mathcal R$ be the local number of <ionizations> per volume per time. <Photoionization equilibrium> makes this equal to the recombination rate. If $\langle E_{\rm rec}(T)\rangle$ is the mean <kinetic energy> removed per recombination, thermal balance reduces to
$$
\mathcal R k_BT_*=\mathcal R\langle E_{\rm rec}(T)\rangle,
\qquad
\boxed{\langle E_{\rm rec}(T_{\rm eq})\rangle=k_BT_*.}
$$
Neither the <ionization> threshold nor the density remains in this event-by-event balance. The threshold energy is stored as atomic <ionization energy> and returned in recombination radiation; it is not a net thermal input. This separation between <photoelectron> heating and <kinetic energy> lost by recombining <electrons> is the relevant cooling balance.<Heating and Cooling of HII regions by Ian Czekala>[Heating by photoionization and Recombination radiation sections]{ref}
In the elementary mean-thermal-energy approximation, a captured <electron> removes $\tfrac32k_BT$. This gives the concise estimate
$$
\boxed{T_{\rm eq}\simeq\frac23T_*
\quad\text{if }\langle E_{\rm rec}\rangle=\tfrac32k_BT.}
$$
That coefficient is a stated approximation, not a universal recombination identity. Real recombination weights the <electron> distribution by its velocity-dependent capture rate. If the chosen cooling model instead writes $\langle E_{\rm rec}\rangle=\eta_{\rm rec}(T)k_BT$, the precise answer is $\eta_{\rm rec}(T_{\rm eq})T_{\rm eq}=T_*$. The source supplies no recombination cooling coefficient or capture kernel from which a unique more accurate coefficient can be computed. Its emitted-photon average is also being used as the average of the absorbed ionizing <photons>; spectral filtering would require a different weighted average.
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