The cheap power supply may actually be current rated properly but its ability to respond may be limited.
Imagine if you will 1 car and a skateboard at the stop light. The car and the skate boarder want to get up to 1 mile an hour, well within the ability of the skateboard or car. When the light turns green the car quickly gets up to one mile an hour while the skateboard requires more time.
The car and skateboard have the potential, Voltage, to go one mile an hour but the car has more ability to provide power (Current) quicker then the skateboard.
I am guessing that the power supply in use does not have the ability to supply the surge current of the WiFi module.