  Static black holes in the conformal anomaly-sourced semi-classical General
Relativity in four dimensions were extended to rotating, stationary solutions,
recently. These quantum-corrected black holes show different features compared
to the Kerr black hole and need for further extensions. Here we remove the
condition of stationarity and find radiating (Kerr-Vaidya-type) solutions in
the same theory augmented with a cosmological constant. As long as the coupling
constant $\alpha$ of the $A$-type trace anomaly is non-zero, we show that $i)$
the cosmological constant is bounded from above, i.e $\Lambda \le \frac{3}{4
\alpha}$; $ii)$ static black holes exist but they may not be unique; $iii)$
static black holes do not satisfy the second law of black hole thermodynamics;
$iv)$ static black holes may have unstable inner horizons; $v)$ In the
nonstationary and axially symmetric case, stability of the event horizon and
the second law of thermodynamics black holes are problematic.
