The phrase "hath a bearing on" refers to something that has a direct impact or relevance to another thing. Synonyms for this phrase include "pertains to," "relates to," "connects to," "impacts," or "influences." These words all convey the idea that one thing is connected in some way to another thing, either by affecting it or being affected by it. Other related synonyms include "is connected to," "has relevance to," "touches upon," and "bears upon." By using these synonyms, you can vary your language and express your ideas with greater precision and clarity, helping to avoid repetition and keep your writing fresh.