  We investigate the effect of spin-lattice coupling (SLC) on the magnetic
properties of the classical kagome-lattice Heisenberg antiferromagnet (KHAF)
using improved Monte Carlo updates. The lattice modes are represented by
Einstein site phonons, which introduce effective further-neighbor spin
interactions in addition to the nearest-neighbor biquadratic interactions. In
the weak SLC, the macroscopically degenerate coplanar ground state remains at
zero field, while a $\sqrt{3} \times \sqrt{3}$ ordered phase accompanied by a
1/3-magnetization plateau appears in external magnetic fields. In the strong
SLC, we find a nematic order at zero field and a 1/9-magnetization plateau
associated with a $3 \times 3$ collinear order. Near the phase transition
between the 1/9- and 1/3-plateau states, the ergodicity in the single spin flip
is practically broken, and slow dynamics appear. We propose that relevant KHAFs
with strong SLC would be realized in spinel-based materials.
