How are electrons in a discharge tube emitted from cathode under low pressure and high voltage ? For example in a thermionic emission they are released due to the excess energy they get as heat which increases their K.E. What's the case in a discharge tube are they emitted from the metallic cathode or are they due to the ionized air?
The high voltage causes the gas molecules in the tube to get excited or ionised if the electric field is strong enough. When the excited electrons go back to a lower energy state, they emit photons in the process. The energy of the photons (and therefore colour of light that we see) depends on the energy of the state in which the excited molecule relative to the energy it has after it is de-excited. The energy to which molecules are excited to depend on the electric field through the gas and therefore you can control the wavelength of the emitted light by adjusting the voltage applied across the tube. For example, if you want more bluish light, you would want to apply a higher voltage than if you want more reddish light (photons corresponding to blue light have higher energy than those corresponding to red light).
Note the above argument is for a Gas Filled Discharge Tube. If you were thinking about a Cathode Ray Tube instead then the process is different. I'm not sure which one you thinking about because your title says cathode and your question says discharge but the links to both are : https://en.wikipedia.org/wiki/Gas-filled_tube