Let us assume a charged particle moving in a circle due to a constant magnetic field. Does this particle emit radiation? If so, where this energy comes from?
Yes, it radiates. The energy of the radiation comes from the kinetic energy of the particle, which would decrease unless some kind of electric field causes the particle not to slow down. The force tending to slow down the particle is called “radiation reaction”. If the particle slows down, the trajectory is a spiral rather than a circle.
For non-relativistic particles, this kind of radiation from circular motion is called cyclotron radiation. For relativistic particles, it is called synchrotron radiation. Calculating the characteristics of this radiation (total power, angular dependence, polarization, frequency dependence, etc.) is a common topic in a graduate-level electromagnetism course.
The radiation is one of the reasons that a particle accelerator like the Large Hadron Collider uses a lot of electricity. The electric power is used to keep the particles from slowing down as they radiate.
Does this particle emit radiation?
Yes. G. Smith explained this in detail in his reply.
If so, where this energy comes from?
The usual answer is that the spin of the particle is responsible for the deflection. If you dig a little deeper, a chain of events becomes visible. The important point here is to understand that electrons are not only charges with an electric field, but also magnetic dipoles (in permanent magnets, the sustained orientation of these dipoles makes up the macroscopic permanent magnetic field).
When the electron enters the external magnetic field, the magnetic dipole of the electron is aligned with the external field. The electron emits electromagnetic radiation and the recoil of the photon causes a) the lateral deflection and b) the disalignment of the magnetic dipole from its alignment to the external magnetic field.
The process repeats as long as the electron has kinetic energy. Exhausting all the kinetic energy by emitting EM radiation, the electron came to standstill in the centre of the spiral. Last not least, the spiral in detail is a sequence of curved pieces, like pieces of tangerine.