Hello there,
I have a bit more different opinion than others I would choose to include the city as part of your keyword, the reason is simple, do a Google search with and without city from the keyword, which one do you think it's more relevant?
Don't forget your longer tail keyword (/w city name) also included your general keyword as well (/wo city name), so there's basically no harm I can think of to include your city name.
But of course, make sure you also use the properly structured data as well.
Some would also suggest that your homepage shouldn't be optimized for keywords, but I don't think there's one-rule-for-all in SEO, especially if you aren't a big brand and just starting out in SEO, your homepage is generally your most linked page with the highest ranking chance. So I would not waste the opportunity to rank keywords with my homepage.