If you want to seriously edit OpenStreetMap while you’re out and about, Vespucci is the way to go. You can add, remove, and modify just about any paths or structures on the map, trace paths using your phone’s GPS and either add them through the app or upload them to use another tool. It uses a lock/unlock system so you don’t make changes accidentally, and you can place geolocated photos on the map to help you position features. It’s easiest to use on a tablet, but still packs its full capabilities into a phone-sized screen.
That said, there’s a steep learning curve. Moving around the map is a bit clunky (which matters less in an editor than it would in a navigator), and the current version is a bit buggy at placing geolocated photos.
Vespucci is available for Android phones and tablets on the Google Play Store, the Amazon App Store, and F-Droid.
Compared to StreetComplete
StreetComplete and Vespucci are complementary. StreetComplete is good for casual mapping (ex: filling in street info), while Vespucci is good for more significant changes (ex: adding a new street). I use both regularly.