Where do electrons come from in thermionic emission? Let we suppose to have a HV Tube or CRT . The filament is connected to the secondary of the transformer used to supply the filament.
At time $t=0$ the wires and cathode are neutrally charged.  Then when we switch the power on, the filament begins to heat and free electrons are being evaporated from the cathode. Now I would come to conclusion that if the electron has left the metal there should remain a hole and the cathode would begin to be more and more positively charged, but this is not happening.
Are those electrons replaced? If yes, then from where? 
Or if they aren't emitted at all, are they generated?

It might be clear from the picture that, electrons emmited by cathode never strikes the anode. The anode is used for accelerating the electron, not to recombine the electron.
Further question is: Where the electrons in anode come from? For accelerating the electron beam, energy is used proportional to the kinetic energy of electrons, but how the current flows if the electons never strike the anode? 
 A: To keep the electrons flowing, you need to connect the anode and cathode of the tube to a suitable circuit.  If the cathode is at a sufficiently negative voltage compared with the anode, then the electrons being emitted by the cathode will be constantly replaced with new ones.
A: Here is an image with the conducting layer all around the glass from the inside:


and here is the circuitry (from a different link):

As you see the circuit closes with the conducting layer. The power supply provides the energy to keep the cathode negative and the anodes positive. In this diagram the  heating of the cathode comes from a different power supply. The electron beam part of the circuit current
Here is another paper

It should be mentioned that the screen has a plate of some metal (often
aluminum) underneath the coating of phosphors; This plate is given a very
strong positive charge, on the magnitude of several thousand volts. This
positive charge pulls the electrons strongly toward the screen.

and of course, close the circuit with the power supply.
With the above in mind,

Now I would come to conclusion that if the electron has left the metal there should remain a hole and the cathode would begin to be more and more positively charged, but this is not happening.
Are those electrons replaced? If yes, then from where?

because the power supply through the circuit replaces the charge

Or if they aren't emitted at all, are they generated?

Of course they are thermally emitted. If there were no thermally emitted beam of electrons the circuit would represent an elaborate capacitor circuit, no current would be flowing.
