Autonomous program improvement typically involves automatically producing bug fixes and feature additions. Such program improvement can be accomplished by a combination of large language model (LLM) and program analysis capabilities, in the form of an LLM agent. Since program repair or program improvement typically requires a specification of intended behavior - specification inference can be useful for producing high quality program patches. In this work, we examine efficient and low-cost workflows for iterative specification inference within an LLM agent. Given a GitHub issue to be resolved in a software project, our goal is to conduct iterative code search accompanied by specification inference - thereby inferring intent from both the project structure and behavior. The intent thus captured is examined by a reviewer agent with the goal of vetting the patches as well as providing a measure of confidence in the vetted patches. Our approach SpecRover (AutoCodeRover-v2) is built on the open-source LLM agent AutoCodeRover. In an evaluation on the full SWE-Bench consisting of 2294 GitHub issues, it shows more than 50% improvement in efficacy over AutoCodeRover. Compared to the open-source agents available, our work shows modest cost ($0.65 per issue) in resolving an average GitHub issue in SWE-Bench lite. The production of explanation by SpecRover allows for a better "signal" to be given to the developer, on when the suggested patches can be accepted with confidence. SpecRover also seeks to demonstrate the continued importance of specification inference in automated program repair, even as program repair technologies enter the LLM era.