I'm not fully understanding cyclotron (and therefore synchrotron) radiation, emitted by a charge in a uniform and constant magnetic field. The charge undergoes centripetal acceleration, therefore the term $a^2$ in the Larmor formula is definitely not zero, thus the power emitted is not zero and we have emission of radiation.
Another true fact is that this magnetic field doesn't do work on the charge, so the kinetic energy of the charge stays the same, in fact it moves on a circumference of constant radius, with constant speed, in the plane perpendicular to the direction of the magnetic field.
The energy emitted doesn't come from the magnetic field cause it is unable to do work on the charge, it doesn't come from a loss of kinetic energy of the charge cause that would make it spiral to the center of the circumference, so Where does the energy emitted comes from?