Let \((M^{n},g,e^{-\phi }dv)\) be a weighted Riemannian manifold evolving by geometric flow \(\frac{\partial g}{\partial t}=2\,h(t),\,\,\,\frac{\partial \phi }{\partial t}=\Delta \phi \). In this paper, we obtain a series of space-time gradient estimates for positive solutions of a parabolic partial equation $$\begin{aligned} (\Delta _{\phi }-\partial _{t})u(x,t)=q(x,t)u^{a+1}(x,t)+p(x,t)A(u(x,t))),\,\,\,\,(x,t)\in M\times [0,T], \end{aligned}$$on a weighted Riemannian manifold under geometric flow. By integrating the gradient estimates, we find the corresponding Harnack inequalities.