  We present two new recovery-based a posteriori error estimates for the
Hellan--Herrmann--Johnson method in Kirchhoff--Love plate theory. The first
error estimator uses a postprocessed deflection and controls the $L^2$ moment
error and the discrete $H^2$ deflection error. The second one controls the
$L^2\times H^1$ total error and utilizes superconvergent postprocessed moment
field and deflection. The effectiveness of the theoretical results is
numerically validated in several experiments.
