Potential energy change in a cyclotron

It is the kinetic energy they're referring to, and not the electrical(and the only) potential energy of the particle, which decreases, right?

Special Relativity four vectors define the kinetics of particles with Lorentz covariant four vectors .

The length of this 4-vector is the rest energy of the particle. The invariance is associated with the fact that the rest mass is the same in any inertial frame of reference.

You ask:

It is the kinetic energy they're referring to, and not the electrical(and the only) potential energy of the particle, which decreases, right?

the kinetic energy is $1/2mv^2$ where the v is the velocity size, momentum is p=mv where both p and v are vectors ,

With each turn the momentum of the particle increases and because of the invariant mass the energy has to increase also. The potential energy in the accelerating fields is irrelevant, it is transferred to the (kinetic energy)/momentum of the particle each cycle in pulses, the last kick at the time it exits. The potential energy in the cyclotron is kept/fed at a calculated synchronous level by the electricity provided.

Particles coming out of the cyclotron have the energy and momentum described above. The potentials entering the Maxwell equations play no role in what is defined as the energy of the beam particles from the cyclotron. ( see this for electromagnetic waves)