Agentic AI has been a topic of great interest recently. A Large Language Model (LLM) agent involves one or more LLMs in the back-end. In the front end, it conducts autonomous decision-making by combining the LLM outputs with results obtained by invoking several external tools. The autonomous interactions with the external environment introduce critical security risks. In this paper, we present a grey-box approach to explore diverse behaviors and uncover security risks in LLM agents. Our approach VeriGrey uses the sequence of tools invoked as a feedback function to drive the testing process. This helps uncover infrequent but dangerous tool invocations that cause unexpected agent behavior. As mutation operators in the testing process, we mutate prompts to design pernicious injection prompts. This is carefully accomplished by linking the task of the agent to an injection task, so that the injection task becomes a necessary step of completing the agent functionality. Comparing our approach with a black-box baseline on the well-known AgentDojo benchmark, VeriGrey achieves 33% additional efficacy in finding indirect prompt injection vulnerabilities with a GPT-4.1 back-end. We also conduct real-world case studies with the widely used coding agent Gemini CLI, and the well-known OpenClaw personal assistant. VeriGrey finds prompts inducing several attack scenarios that could not be identified by black-box approaches. In OpenClaw, by constructing a conversation agent which employs mutational fuzz testing as needed, VeriGrey is able to discover malicious skill variants from 10 malicious skills (with 10/10= 100% success rate on the Kimi-K2.5 LLM backend, and 9/10= 90% success rate on Opus 4.6 LLM backend). This demonstrates the value of a dynamic approach like VeriGrey to test agents, and to eventually lead to an agent assurance framework.