You're right that you do not want two pages competing for the same words. It will dilute your efforts.
It might help to reiterate why homepages generally rank better than others - it's not because there's something intrinsic about a homepage that makes it stronger than others - it's because other than with a few rare exceptions, homepages receive the bulk of internal and external links.
This makes ranking an inner page for a competitive keyword that much more difficult, but certainly not impossible. It's actually straightforward to make an inner page the "strongest" page on a site by linking internally in unusual ways. But this is usually undesirable - even if you want a strong inner page, you almost always want your homepage to be the "default" entry point to your site.
Depending on the competition for your keyword, the main factor to consider is your ability to get high value links to your target URL. If the competition for the keywords is a bunch of homepages, the you are going to have a lot of ground to catch up.