flavios:
The potentiometer (10k) used for contrast regulation is only working at the very end of its top span.
Actually, its bottom span - the end connected to ground. This is because of a long-standing and stupid foul-up in the design which is a mistake made many years ago and mindlessly perpetuated from one published circuit to another. Not the only time such a thing has happened. 
As photoncatcher has attempted to explain, the potentiometer should only connect with the wiper to pin 3 ("Vo") and one end to ground, the other end should not be connected, or certainly not to 5 V. And the value if you want better control should be 1k, not 10k but if you are using a 10k, a clever trick to make the best of a bad situation is to connect both ends to ground.
flavios:
I tried to put another resistor in series (4.7k). My intention was to extend the range or, more precisely, to move it up (4,7k to 14.7k). But it does not work.
It most certainly will not, because the problem here is far too much resistance, not too little.
photoncatcher:
The background illuminator LED receives power via pin 15 - there must be a 220 ohm resistor in series.
In most of the 1602 displays available, Resistor R8 on the board is "101" or 100 Ohms and a resistor in series is entirely unnecessary, unless you wish to dim the display.
photoncatcher:
An alternative cause might be that your Arduino is performing power-hungry calculations
Complete nonsense! 
photoncatcher:
or is powering lots of auxiliary stuff - servos, lots of LEDs, etc..
And that certainly may be a problem.
You should not be attempting to power a system via the "barrel jack" or "Vin" as the on-board regulator is only capable of supplying the on-board chips and essentially, nothing else beyond a coupe of LEDs. 