My home page ranking higher than my optimized page for a keyword

I optimized my homepage for “cleaning services city” but saw a lot of traffic on another keyword “house cleaning city”. So I decided to create a page optimizing for that keyword. Headings, meta data, etc. but my homepage is beating that page by 2 positions. I’m afraid of cannibalization. I’m thinking of building links for my “house cleaning city” page, since my home page may be ranking higher due to it having more authority from links, and still being highly relevant to the keyword. Let me know what you guys think.