In this study, we investigate Hinglish—a blend of Hindi and English commonly found in informal online communication—with a particular focus on automated pun generation. Our work examines the applicability and adaptability of existing English pun generation pipelines to Hinglish. We assess the pun generation capabilities of Large Language Models (LLMs), particularly GPT-3.5. By employing Chain of Thought prompting and Self-Refine techniques, we identify cross-linguistic homophone detection as a central difficulty. To address this, we propose a novel algorithm for cross-lingual homophone identification and develop a Latin-to-Devanagari transliteration module to leverage the widespread use of Latin-script Hindi in online settings. Building on existing frameworks for pun generation, we incorporate our homophone and transliteration modules to improve output quality. Crowd-sourced human evaluations validate the effectiveness of our approach.