The power of a laser is controlled to some extent by the power supply. Unless there is some feedback mechanism to change the output of the laser during cutting, the output of the laser remains constant (in mechanical cutting, the resistance of the material to cutting is mechanically communicated to the power source, generally a motor, which draws more current in response to thicker/tougher sections of material). In the case of a laser, it supplies a surplus of power to cut/ablate the material; if the remaining laser power could be measured, (for example, by a detector beyond the material being cut) then the power used in cutting could be measured by difference.
Yes Laser power input to the workpiece genally varying during cutting because the absorption coefficients of materials is varying. It can be measured by inverse analysis by measuring the thermal history of the material.
There are two different values: output power of laser and power, which is absorbable by the material being cut. Output power of laser usually is constant. Absorbable power usually significally decrease after the melted layer forms on the material surface. It is due to decrease the light absorbtion coefficient.
To measure this decrease you can measure the rate of the material heating near an area of the laser influence.