The question basically is: is there a generator that utilizes external electric field in addition to conventional magnetic field.
Since that is not a concept that is routinely covered in text books, the question must be whether or not the concept is practicable or has ever been explored or proven. Since this is actually two concepts, one of which appears to be the dual of the other, it might be useful to consider the combination of two machines. Since the electrostatic machine would tend to operate at a high voltage and low current while the electromagnetic machine would tend to operate at a low voltage and high current they could not easily be connected together electrically. That would tend to indicate that the two concepts can not usefully be combined in one machine.
There are many electronic power conversion products available for creating electrical grids and microgrids that combine a variety of electric power generating sources. Combining an electrostatic machine with an electromagnetic machine in that manner is certainly practicable.
Some recent macro scale electrostatic machine research is described by: B. Ge; A. N. Ghule; D. C. Ludois, "A 3D Printed Fluid Filled Electrostatic Rotating Machine Designed with Conformal Mapping Methods," in IEEE Transactions on Industry Applications, 09 May 2017 Vol. 52(3), pp. 2156-2166.
The above paper identifies 13 electrostatic machines that were constructed between 1933 and 2016 and provides references to the associated research. The list consists of ten macro-scale machines and three micro-scale machines. The list includes the speed, torque, voltage and design concept for each machine.