In this paper, we obtain some gradient estimates for positive solutions to the following nonlinear parabolic equation $$frac{{partial}u}{{partial}t}={ riangle}u-b(x,t)u^{sigma}$$ under general geometric flow on complete noncompact manifolds, where 0 < ${sigma}$ < 1 is a real constant and $b(x,t)$ is a function which is $C^2$ in the $x$-variable and $C^1$ in the$t$-variable. As an application, we get an interesting Harnack inequality.