Large Language Models (LLMs) are increasingly embedded in software via APIs like OpenAI, offering powerful AI features without heavy infrastructure. Yet these integrations bring their own form of self-admitted technical debt (SATD). In this paper, we present the first large-scale empirical study of LLM-specific SATD: its origins, prevalence, and mitigation strategies. By analyzing 93,142 Python files across major LLM APIs, we found that 54.49% of SATD instances stem from OpenAI integrations and 12.35% from LangChain use. Prompt design emerged as the primary source of LLM-specific SATD, with 6.61% of debt related to prompt configuration and optimization issues, followed by hyperparameter tuning and LLM-framework integration. We further explored which prompt techniques attract the most debt, revealing that instruction-based prompts (38.60%) and few-shot prompts (18.13%) are particularly vulnerable due to their dependence on instruction clarity and example quality. Finally, we release a comprehensive SATD dataset to support reproducibility and offer practical guidance for managing technical debt in LLM-powered systems.