The increasing availability of interventional data offers new opportunities for causal discovery, with gene perturbation studies providing a prominent example. Such data are typically count-valued and subject to substantial measurement error arising from technical variability and latent state heterogeneity. Motivated by these challenges, we study identification and estimation in latent linear structural causal models for interventional count data. We propose a latent linear Gaussian directed acyclic graph (DAG) model with Poisson measurement error that explicitly separates the latent causal structure from the observed counts. Under one-sparse mechanism shifts, allowing the structural coefficients and intercept of the targeted gene to change, we establish population-level identifiability of the full latent causal DAG when every node is covered by at least one non-vanishing targeted intervention. Building on these identification results, we develop an estimation procedure based on sparse inverse matrix estimation and provide theoretical guarantees on estimation error and finite-sample causal discovery. Simulation studies and applications to Perturb-seq data demonstrate the practical effectiveness of the proposed method.

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