electromagnetism - Confusion about Cyclotron (and synchrotron) radiation

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?

Answer

The energy emitted doesn't come from the magnetic field cause it is unable to do work on the charge,...

You are right about that the magnetic field in sum doesn't contribute any work to the moving and by this deflected electron. The mechanism is more subtle:

1. It is well known that the electron has not only a charge but also a magnetic dipole moment. This moment - in contrast to the electric charge - has an orientation. This north-south orientation is equally distributed in space for the incoming electrons.


2. The cyclotrons magnetic field align the magnetic dipole moments of the moving electrons. At the moment of the alignment a tiny amount of the kinetic energy of the moving electron gets emitted. The electron radiates a photon.


3. 
   

   The emission of a photons is accompanied with a momentum transfer. This transfer has two consequences for the electron. The electrons magnetic dipole moment gets disaligned again and the electron gets shifted in one direction (which states the Lorentz force).

   
   

   3.1 A comment. The magnetic dipole moment of subatomic particles is connected with the intrinsic (means every time existing property) spin. Both have a direction and if this directions are defined as parallel for electrons then the directions for the positron is antiparallel. The introduction of spin is due to the fact that if the electron in the cyclotron is deflected to the right then the positron will be deflected to the left.

   
   
   

... 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

As it was explained above the radiation of the charge consumes kinetic energy and the charge indeed spiraling to a center. To be precise it goes in tangerine slices step by step along a spiral path until its kinetic energy is exhausted.

Funny is by the way the imagination that the photons which push the electron into its movement riding biggypack on the electron and get shaked off during the electrons deflection inside a magnetic field.

Another true fact is that this magnetic field doesn't do work on the charge,...

This statement is true from the statistical point of view. But in detail the magnetic field acts like a spring acts. Both are able to accumulate energy. In the case of the cyclotron the alignment and later the disalignment of the electrons magnetic dipole moment makes the external magnetic field sometimes weaker and sometimes stronger but after the electron leave the field it stays unchanged.