Precision Medicine (PM) transforms the traditional "one-drug-fits-all" paradigm by customising treatments based on individual characteristics, and is an emerging topic for HCI research on digital health. A key element of PM, the Polygenic Risk Score (PRS), uses genetic data to predict an individual's disease risk. Despite its potential, PRS faces barriers to adoption, such as data inclusivity, psychological impact, and public trust. We conducted a mixed-methods study to explore how people perceive PRS, formed of surveys (n=254) and interviews (n=11) with UK-based participants. The interviews were supplemented by interactive storyboards with the ContraVision technique to provoke deeper reflection and discussion. We identified ten key barriers and five themes to PRS adoption and proposed design implications for a responsible PRS framework. To address the complexities of PRS and enhance broader PM practices, we introduce the term Human-Precision Medicine Interaction (HPMI), which integrates, adapts, and extends HCI approaches to better meet these challenges.