A new carbon-based mono-layer atomic crystal, phagraphene, was proposed recently, which has received increasing attention. We here consider variations of edge chemistry for a phagraphene nanoribbon, and also explore to expand its functional properties, some typical nonmetallic atoms are used as ribbon-edge terminations. With these edge chemistry, the bandgap of ribbons may be nearly unchanged, increased, or decreased as compared with the bare-edge ribbon, and even a quasi-metal or metal arises, presenting rich and flexibly tunable electronic structures. For these features, new emerging hybridized state subbands upon edge terminations between two intrinsic subbands play an important role. In particular, by several types of termination atoms, ribbons exhibit significantly enhanced carrier mobility, and diverse edge terminations can effectively control the carrier mobility to a difference of three orders of magnitude, which can be used to explain the fact that the various measured mobility is quite different for the same nanomaterial when synthesized by different ways. Furthermore, the introduction of non-hydrogenation terminations can substantially improve the electronic transport features of ribbon, such as a appearance of impressive negative differential resistance phenomena. These studies suggest that resulting structures presented here might possess promising applications in future electronics.