Do photons have kinetic energy?

Yes. The relativistic definition of kinetic energy $$K$$ for a particle of mass $$m$$ is

$$K=E-mc^2=\sqrt{(mc^2)^2+(pc)^2}-mc^2\approx\frac{p^2}{2m}+…$$

where $$E$$ is the relativistic energy and $$p$$ the relativistic momentum.

Set $$m=0$$ and you get

$$K=E=pc$$

for a photon. This relation involves only Special Relativity.

In addition, quantum mechanics tells us that the energy is related to the angular frequency $$\omega$$ by

$$E=\hbar\omega$$

and the momentum is related to the wavenumber $$k$$ by

$$p=\hbar k$$

so we get the expected relation between angular frequency and wavenumber for an electromagnetic wave,

$$\omega=kc.$$

The photons of a radio wave and a gamma wave have different frequencies and thus different energies, and also different wavenumbers and thus different momenta. They may have the same speed $$c$$ but they have different $$\omega$$, $$k$$, $$E$$, and $$p$$ and this makes them interact differently with other particles.

It is called special relativity, and it is the kinematics ruling at the level of particles and of large velocities generaly, close to the speed of light.

The concept of Energy in special relativity includes the energy inherent in the rest mass of the system .

$$\sqrt{P\cdot P}=\sqrt{E^2-(pc)^2}=m_0c^2$$

Here p is the momentum vector of the particle, and one can say the $$(pc)$$ is the kinetic energy term of the particle in special relativity. When mass equals zero, as with the photon, the total energy is kinetic energy. For photons the $$E=hν$$ holds, where $$h$$ is Planck's constant and $$ν$$ the frequency of light.

Thus, it is the difference in the frequency that differentiates a gamma ray photon and a radio wave photon.

Photons behave a little like mechanical objects and a little like their own thing in this regard.

Suppose you have a squirt gun that shoots a series of water droplets. If you sit still with respect to the gun, the droplets hit you with a certain frequency, momentum, and energy. If you run toward the gun, the frequency, momentum, and energy all go up.

The energy of a photon is proportional to its frequency. $$E = h \nu$$.

If you run upstream into a beam of light. The frequency increases because of the Doppler shift. And so do the momentum and energy of the photons. If you ran at a suitable relativistic velocity, you could turn a radio wave into a gamma ray wave.

At the same time, you do not increase the speed the photons travel with respect to you by running. They always travel at the speed of light.