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The combined cooling, heating and power (CCHP) system is a typical distributed, electricity-gas integrated energy scheme in a community. First, it generates electricity by use of gas, and then exploits the waste heat to supply community with heat and cooling. In this paper, we consider a smart city consisting of a number of communities (CCHPs) and an agent of power grid (APG), where CCHPs can sell energy to the APG according to its bid. To study all utilities of entities in such a city from energy trading, a noncooperative Stackelberg game between APG and CCHPs is formulated. Here, the APG gives a bid for buying the energy from CCHPs, then CCHPs respond to the APG with their optimal energy supply that maximizing their utilities according to this bid. We show that the maximum profit to the APG and utilities to the CCHPs can be obtained at the Stackelberg equilibrium, which is guaranteed to exist and unique. Because the complete information about energy supply of each CCHP is unknown to the APG in advance, we propose a distributed algorithm that is able to find the point of equilibrium through a limited number of iterations. Taking privacy protection and transaction security into consideration, we design a blockchain-enabled energy management system. This system is composed of Internet of Energy (IoE) sub-system and blockchain sub-system, where the information interactions as well as energy transactions between APG and CCHPs can be carried out effectively and safely. Finally, security analysis and numerical simulations show the effectiveness and accuracy of our proposed mechanism.