Cross-lingual summarization refers to generating a summary in a target language different from the source document. Currently, most work in this field simplistically constructs training samples for cross-lingual summarization by translating mono-lingual summarization corpora, overlooking the potential hallucinations and errors this may introduce. Furthermore, existing cross-lingual summarization methods typically fail to address the enhancement of factual consistency in generated summaries, resulting in outputs with low factual accuracy. To address these limitations, we propose a Factuality Optimization based on Auxiliary Information method (FOAI) for cross-lingual summarization. FOAI first constructs a cross-lingual summarization corpus with reduced hallucination and error rates using a context-guided approach. Subsequently, during the fine-tuning stage, a factuality optimization mechanism is introduced. This mechanism incorporates an auxiliary module that generates factual signals, augmenting the input to the large language model (LLM) to effectively enhance the factual consistency of the generated cross-lingual summaries. Experimental results on cross-lingual datasets derived from CNN/DailyMai and XSum demonstrate that FOAI successfully introduces a factual optimization mechanism into cross-lingual summarization tasks. The study further analyzes the impact of different data scales on experimental outcomes.