Hi,
Did you measure the INPUT VOLTAGE to the booster when the problem occurs.
Don't forget power in will be higher than power out, and as you are going from 3.8V under load and expecting 5V out with current of 500mA.
Thats 5 * 0.5 = 2.5Watts out.
96% efficiency, they claim....
2.5* (100/96) = 2.6W input power
InputCurrent = 2.6 / 3.8 = 0.684Amps
So probably 0.75A would be a possibility for input current considering its from "worldchips" china.
The other thing to consider is it built to manufactures suggested circuit, or to price?
0.9V input is a bit suspect for a start.
Have you tried putting a 1000uF 16V electro cap across the input to the booster?
@mauried suggestion is the best of all, I have quite a few of these for 5V supplies.

Tom..