Hands are absolutely an infection route, by acting as a transport vehicle to your nose and eyes. Gloves don't change that fact though, unless it reminds you to not touch your face. (It does to me: I work with gloves and epoxy resin a lot, and I've developed a pretty strong reflex to not touch myself with gloved hands even though I do it all the time with bare hands.)
It was on NBC news two nights ago that the virus does not infect you through eyes or blood. It can only infect and replicate in the lungs. So only nose and mouth if that report was correct.
Several states fairly recently passed laws requiring food preparers to wear gloves. Then they backtracked when they realized that they actually decreased safety.