We extend the theoretical framework used to study search processes with stochastic resetting to the case of partially absorbing targets. Instead of an absorption event occurring when the search particle reaches the boundary of a target, the particle can diffuse freely in and out of the target region and is absorbed at a rate κ when inside the target. In the context of cell biology, the target could represent a chemically reactive substrate within a cell or a region where a particle can be offloaded onto a nearby compartment. We apply this framework to a partially absorbing interval and to spherically symmetric targets in ${\mathbb{R}}^{d}$. In each case, we determine how the mean first passage time (MFPT) for absorption depends on κ, the resetting rate r, and the target geometry. For the given examples, we find that the MFPT is a monotonically decreasing function of κ, whereas it is a unimodal function of r with a unique minimum at an optimal resetting rate ropt. The variation of ropt with κ depends on the spatial dimension d, decreasing in sensitivity as d increases. For finite κ, ropt is a non-trivial function of the target size and distance between the target and the reset point. We also show how our results converge to those obtained previously for problems with totally absorbing targets and similar geometries when the absorption rate becomes infinite. Finally, we generalize the theory to take into account an extended chemical reaction scheme within a target.