  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 $\kappa$ 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 $\R^d$. In each case, we determine how the mean first
passage time (MFPT) for absorption depends on $\kappa$, the resetting rate $r$,
and the target geometry. For the given examples, we find that the MFPT is a
monotonically decreasing function of $\kappa$, whereas it is a unimodal
function of $r$ with a unique minimum at an optimal resetting rate $r_{\rm
opt}$. The variation of $r_{\rm opt}$ with $\kappa$ depends on the spatial
dimension $d$, decreasing in sensitivity as $d$ increases. For finite $\kappa$,
$ r_{\rm opt}$ 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.
