Misinformation about climate change causes numerous negative impacts, necessitating corrective responses. Psychological research has offered various strategies for reducing the influence of climate misinformation, such as the fact-myth-fallacy-fact-structure. However, practically implementing corrective interventions at scale represents a challenge. Automatic detection and correction of misinformation offers a solution to the misinformation problem. This study documents the development of large language models that accept as input a climate myth and produce a debunking that adheres to the fact-myth-fallacy-fact (“truth sandwich”) structure, by incorporating contrarian claim classification and fallacy detection into an LLM prompting framework. We combine open (Mixtral, Palm2) and proprietary (GPT-4) LLMs with prompting strategies of varying complexity. Experiments reveal promising performance of GPT-4 and Mixtral if combined with structured prompts. We identify specific challenges of debunking generation and human evaluation, and map out avenues for future work. We release a dataset of high-quality truth-sandwich debunkings, source code and a demo of the debunking system.