The total baryonic mass contained in galaxy clusters are known to be far less than that required to gravitationally sustain the measured motions of the individual galaxies in the cluster, as per Newtonian gravity. In order to explain the discrepancy, it has been proposed and long believed that there is significant amount of “dark matter” contained in the galaxy clusters. Similar dark-matter hypothesis has also been invoked to explain fast rotation in spiral galaxies. Modification of the laws of Newtonian gravity or dynamics has been proposed as an alternative solution to the dark-matter hypothesis. However, success of such modified theories is primarily limited to explain fast rotation in spiral galaxies, but have remained generally unsatisfactory to explain mass discrepancy in galaxy clusters. Besides the discrepancy in the total mass, the required distribution of the total mass from gravitational lensing measurement, particularly for the specific Bullet Cluster, have posed a critical challenge for the modified gravity/dynamics theories. A new modified-gravity theory, referred to as a unified electro-gravity (UEG) theory, which introduces a new gravitational field in the presence of an electromagnetic field or radiation, has been proposed and successfully applied to model elementary charged particles such as an electron, as well as the flat rotation curves of spiral galaxies, the results of which were presented in the last AAS conference. We present here the application of the UEG theory to model the total mass as well as the mass distribution in galaxy clusters. The UEG theory, properly applied for a specific distribution of baryonic mass in a galaxy cluster, in the presence of the cosmic microwave background radiation, results in a gravitational field which effectively substitutes for the required total mass (gravitational) and its distribution in a galaxy cluster. A parameter constant of proportionality between the UEG field and the local effective energy density of microwave radiation, required in the new model, is borrowed from the UEG model of an electron which is known to be consistent with that of a spiral galaxy. Results from the new UEG theory, as applied to galaxy clusters in general, as well as specifically applied to Virgo and Bullet clusters, are presented and compared with existing measurements. For the Bullet cluster, the UEG theory predicts that the maximum gravitational mass is concentrated at locations that are radially somewhat offset away from the center, as compared to the baryonic mass concentrations in the cluster. This is shown to be consistent with gravitational lensing measurement.