What is the name and purpose of the following component?

I was wondering what is the name and purpose of the white box component ? It seems to be just a wire with some white body around. Is it a resistor or a fuse ?

It's a resistor, perhaps wirewound.

Look at the PCB where that component is soldered. Its component designator will probably be a number prefixed by "R". In combination with that capacitor C25, this is a "capacitive dropper" from the incoming mains voltage.

You can read more about that type of power supply in this previous question and elsewhere.

My guess is that 4W22R means 4 watts 22 ohms

Yes.

How can I test that component is still working ?

With power completely removed and the capacitors on the PCB discharged, you could try measuring its resistance with a DMM and see if you get a sensible result. Since I expect it to be in series with C25 and no other components in parallel with it, then you should measure 22 ohms +/- 10% 5%

The lack of discolouring gives us no indication of overheating (although doesn't disprove that it has occurred).

Its a resistor of 4 W having 22 Ω resistance and the letter J means a tolerance of + or - 5%.

A multimeter will give its resistance and if it's faulty it will give no resistance.

The capacitor and the resistor drop the voltage (transformerless power supply) before it's converted to DC by diodes.

22R 4W as already established

WARNING This sort of power supply can kill you many seconds before you realise that you are dead.

ALL parts of the whole circuit MUST be treated as if they are art mains potential when mains connected as sometimes they may be.

Zener failure indicates that the circuit was "designed" about as badly as such circuits usually are and as all should be considered to be. They may die due to mains surge, component under-rating or Murphy, and 'other things'.

Had the zener failed o/c, as happens less often than s/c, but often enough, modified mains voltage would have been applied elsewhere and das spitzen sparkem, bad smelzen and flashing lights maybe. Briefly.