The purpose of using a centre-tap return connection on the secondary of transmitting tube's filament transformer is to:

In many high-power transmitting vacuum tubes, the filament also acts as the cathode (a directly heated cathode). If it is heated with alternating current (AC), the voltage variations of the AC supply can cause the emitted electron stream to fluctuate at the AC frequency, introducing an unwanted 60 Hz or 120 Hz hum into the transmitted signal. To prevent this, a centre-tap on the filament transformer's secondary winding is used as the return point for the high-voltage plate circuit. By returning the grid and plate circuits to this electrical centre, the AC voltage swings on either side of the filament balance each other out, canceling the hum. Distractors suggesting it keeps voltage constant or reduces harmonics are incorrect; the centre-tap's sole purpose in this context is to provide a balanced ground reference to eliminate AC hum modulation.