Although there have been plenty of reports about graphene transistors and photodetectors, due to the carbonization of photoresists during Reactive Ion Etching (RIE) process, polymer residue is widely found in graphene devices, which leads to a weak photoresponse and low carrier mobility. On the other hand, doping in ambient conditions also result in mobility deterioration and Dirac point shift in graphene devices. In this paper, a novel lithography process is found to be efficient in alleviating the polymer residue problem. By inserting a diluted Polymethyl methacrylate (PMMA) interlayer between graphene and photoresists, and using sodium hydroxide (NaOH) solution for strapping, the photoresponse of pure monolayer graphene can be as large as 220 mA/W at 1 V bias, which is much larger than previous reports. Furthermore, by introducing an ammonia solution process during the graphene transfer process, the maximum hole mobility of the back-gated graphene transistor can be larger than 8500 cm(2)V(-1)s(-1), with the Dirac point changed to be near zero. More practically, the mobility exhibits a good stability, which remains high after 20 days in ambient condition. This work provides a feasible route for improving the photoresponsivity and carrier mobility of graphene-based photodetectors.