If an electron which already possesses some kinetic energy of $X\rm\,eV$ is further accelerated through a potential difference of $X′\rm\,V$, then is it correct to state that the electron now has a total kinetic energy of $(X+X′)\rm\,eV$? Using this information, am I allowed to substitute this value in the de Broglie equation to find the wavelength of the electron?
Yes this is always true.
Bear in mind that if the electron energy is comparable to its rest mass you will need to use the relativistic expression linking its energy and momentum:
$$ E^2 = p^2c^2 + m^2c^4 $$
But apart from this proceed as you describe.