At DESY, the development of a 1.6-cell, 1.3 GHz all superconducting gun cavity with a lead cathode attached to its back wall is ongoing. The special features of the structure like the back wall of the half-cell and cathode hole require adaptations of the procedures used for the treatment of nine-cell TESLA cavities. Unsatisfactory test results of two prototype cavities motivated us to re-consider the back-wall design and production steps. In this contribution we present the status of the modified cavity design including accessories causing accelerating field asymmetries, like a pick up antenna located at the back wall and fundamental power- and HOM couplers. Additionally, we discuss preliminary considerations for the compensation of kicks caused by these components.