  We derive Harnack inequalities for a stochastic reaction-diffusion equation
with dissipative drift driven by additive rough noise in the $L^p$-space, for
any $p \ge 2$. These inequalities are used to study the ergodicity of the
corresponding Markov semigroup $(P_t)_{t \ge 0}$. The main ingredient of our
method is a coupling by the change of measure. Applying our results to the
stochastic reaction-diffusion equation with a super-linear growth drift having
a negative leading coefficient, perturbed by a Lipschitz term, indicates that
$(P_t)_{t \ge 0}$ possesses a unique and thus ergodic invariant measure in
$L^p$ for all $p \ge 2$, which is independent of the Lipschitz term.
