Is it working just with bare Esp32 board (without RFID DF etc) powered from TP?
without MT3608. 5V from powerbank directly to 5V pin.
with MT3608 -> have always 5.2V at the MT3608 pins, and ~5.0-5.1V at the 5V pin of my controller.
without RFID and DFPlayer but WITH MT3608 and TP4056 - no, not working.
without MT3608 and TP4056, and without RFID - works.
so, once again.
i could say, it only works, if I give 5V power supply directly to my 5V pin on the controller, WITHOUT my RFID and with DFPlayer. But in 1-2 min it make reset and after not always starting.
If you install the battery, then plug external power into the TP4056 module for about one second, then remove it, does the battery then power the circuit properly? If not, then it appears the boost converter isn't working properly, or can't provide enough current, or your battery isn't able to provide enough current.
Even so, as others have said, you should not be passing load current through the charger. You need a "load sharing" circuit, which includes a P-channel mosfet, a shottky diode, and a resistor.
During that 1-2min you could measure the real supply voltage.
The player needs between roughly 4 V and a max of 5.0 V. Until you have it playing satisfactorily, keep the volume setting to 20. At its max of 30 current consumption can cause troublesome voltage drops. I rarely use over 25.
Of course, it depends on the volume of the original recording!
Hello together, i have solved this problem adding 220uF capacitor at the MT3608 output.
Thank you all.
I was under the impression you were using a module ?! a module would have a 22uF capacitor on it. That should normally suffice. Also the ESP32 will have an input voltage capacitor on it.
I just said what helps 
P.S. manufacture has also confirm it!
Have a nice day