Large Language Models (LLMs) have improved performance across various natural language processing tasks. Despite these improvements, LLMs continue to face significant challenges, such as grammatical issues and code-switching to English, when applied to low-resource languages like Cantonese in Machine Translation (MT) scenarios. By addressing the unique linguistic and contextual challenges of Cantonese, we present a novel strategy to improve the understanding and translation capabilities of LLMs for Cantonese-to-Mandarin MT. Our strategy comprises three key components: (1) Syntax and Part-of-Speech (POS) fine-tuning, where we use the Universal Dependencies (UD) corpus to fine-tune LLM, focusing on the linguistic structures of Cantonese; (2) Specialized Cantonese to Mandarin sentence pairs, collected from diverse sources such as Cantonese grammar textbooks and manually translated sentences across various domains, to expose the model to a wide range of linguistic contexts; (3) Post-processing with additional LLMs, where we introduce additional LLMs to improve the initial translations, correcting Mandarin grammar and punctuation. Empirical evaluations on human-created test sets show that our proposed strategy improves translation performance and outperforms existing commercial translation models with at least 3 BLEU scores. Additionally, our strategy also benefits other LLMs and a reversed translation direction, demonstrating its generalization and effectiveness.