We study a one-dimensional diffusive particle subject to stochastic resetting to its initial position, in the presence of an imperfect, localized target that can absorb (kill) the particle, modeled by a delta-function killing rate. The central question is how stochastic resetting competes with drift-induced transience and the target's finite reactivity. Exact Laplace-space expressions are derived for the non-normalized propagator and survival probability, for arbitrary diffusion coefficient $D$, resetting rate $r$, and killing strength $k$, first in the absence of drift and then under a constant drift. The long-time behavior separates into distinct regimes. Without resetting, unbiased diffusion exhibits the recurrent algebraic survival law $S\left(t\right)\sim t^{-1/2}$, whereas any nonzero drift renders the motion transient with respect to the target and leaves a nonzero ultimate survival probability. By contrast, any $r>0$ and $k>0$ repeatedly renew encounters with the target and restore eventual absorption, yielding an exponential survival law $S\left(t\right)\sim A{e}^{-\theta t}$. The decay rate $\theta$ is given by the dominant (rightmost) pole of the Laplace transform, and the surviving density converges after normalization to an explicit quasistationary profile. The driftless and perfectly absorbing limits recover standard resetting results. These formulas provide a unified description of the crossover between recurrence-controlled, transience-controlled, and renewal-controlled survival.

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• Brownian motion
• Classical statistical mechanics
• Diffusion
• Nonequilibrium statistical mechanics
• Stochastic processes
• Stochastic resetting