Conservation of energy, incident and emitted radiation Suppose we have an electromagnetic wave the propagates through space,  in the direction of x-axis. A charged particles lies along the x-axis and at some time $t_0$ the EM wave hits the particle. The electric field exerts a force:
$$\mathbf{F} = \frac{\mathbf{E}}{q}$$
I can't understand the way energy is conserved in this case and the interplay between incident and emitted radiation (we know that light must be emitted because the charged particle is accelerated by the force).
The EM wave before time $t_0$ has an energy $E_{inc}$, that is the energy of the incident wave. When it hits the atom it loses some energy which is converted in kinetic energy of the particle. So at this point we have:
$$E = K + E'_{incident} + E_{emitted}$$
where $E'_{incident}$ denotes the energy of the original wave after interaction with the particle has take place. Is this description correct? I tried to find some classical description of such situation and during searching I found the following:

Optical radiation accelerates charges in a
material, and accelerating charges emit light that adds to (or subtracts from) the incident
light.

I can't understand the final statement, that is how emitted light adds or subtracts from the incident light if the energy is conserved?
 A: 
The EM wave before time $t_0$ has an energy $E_{inc}$, that is the energy of the incident wave. When it hits the atom it loses some energy which is converted in kinetic energy of the particle. So at this point we have:
$$E = K + E'_{incident} + E_{emitted}$$
where $E'_{incident}$ denotes the energy of the original wave after interaction with the particle has take place. Is this description correct?

No.
We can't assign separate energies to "partial waves" and add them up.
In macroscopic EM theory, EM energy is associated with any space region in which there are EM fields, but this energy can't be distributed/assigned in some objective and useful way to individual elementary waves present.
One correct description of the process would be: when the EM wave hits a charged particle initially at rest, the particle accelerates and thus produces secondary wave spreading out from the particle.
In terms of energy, some EM energy (in the space around the particle, due to EM field of both primary wave and secondary wave being present) is being transferred to the particle and this shows up as increased kinetic energy or internal potential energy of the particle ( if the particle has inner degrees of freedom). Some other EM energy is being redirected in direction; before the interaction, EM energy moved in direction of the primary wave, but when interaction is going on, a different energy flow appears, where some energy is redirected to all possible directions (due to presence of the secondary wave).

Optical radiation accelerates charges in a material, and accelerating charges emit light that adds to (or subtracts from) the incident light.
-- I can't understand the final statement, that is how emitted light adds or subtracts from the incident light if the energy is conserved?

They mean EM field. Accelerating charge produces its own EM field, which adds up to the field of the primary wave. This does not mean "energies of secondary and primary wave" add up. There are no such separate energies. There is only total EM energy of the resulting EM field.
A: The total energy is $E_{incident+emmitted} + E_{mechanical}$, the mechanism by which the "incident" waves energy is reduced, is due to the superposition of the emitted wave and the incident wave. The incident wave doesn't change to ', it stays as it is, and the resultant field energy  of the 2 waves decreases.
The "lost" field energy, is made up for in the increase in mechanical energy.
Field energy is not a conserved qauntity, total energy is.
