Do photons have kinetic energy? Do photons have kinetic energy?
If a photon of radio wave and a photon of gamma wave moves at the same speed, how can both of them have different energies?
 A: 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. 
A: It is very confusing when you read the definition of kinetic energy (the energy of a particle that it possesses due to its motion), and since both photons (with different wavelength) travel at the same speed c, you could think they should have the same kinetic energy.

It is defined as the work needed to accelerate a body of a given mass from rest to its stated velocity. Having gained this energy during its acceleration, the body maintains this kinetic energy unless its speed changes. The same amount of work is done by the body when decelerating from its current speed to a state of rest.

https://en.wikipedia.org/wiki/Kinetic_energy
But the photon is massless, and it does not have a rest frame, it by definition travels at speed c.
Photons always travel at speed c in vacuum, when measured locally. Its energy and momentum are related by E=p*c (where p is the magnitude of the momentum vector).
$E^{2}=p^{2} c^{2} + m^{2} c^{4}$
Since the photon is massless, this will reduce to E=pc.
The energy and momentum of the photon only depend on its frequency or inversely on its wavelength. $E=h\nu=\frac{hc}{\lambda}$
$p=\frac{h\nu}{c}=\frac{h}{\lambda}$
The photons energy (E) is kinetic energy (p*c) and in your case the different wavelength photons have different energy and momentum. Still, they both travel at speed c in vacuum when measured locally.
A: Yes.
Moreover, you could say the energy of a photon is purely kinetic energy.
In relativity theory, massive particles have both kinetic energy and a potential energy which is proportional to their mass. Photons have no mass, hence their energy is purely, and wholly, kinetic.
A: 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.
A: 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. 
A: massless things can never have inertia, nor momentum.
ki·net·ic en·er·gy
/kəˈnedik ˈenərjē/
nounPhysics
noun: kinetic energy; plural noun: kinetic energies
energy which a body possesses by virtue of being in motion.

you need MASS to have inertia, light i.e. photons have none.
however i can still answer the part about why "a radio wave and a photon of gamma wave moves at the same speed, how can both of them have different energies?"
the answer is that the higher the frequency you go with electromagnetic waves, the more energy they have.
for example light that is red will have less energy than light that is blue. and ultraviolet light will have even more !
gamma waves are at the highest level that have been detected by mankind. they are exceedingly high in frequency.
