I have circuit which is connected in the breadboard containing components PIC16F877A, ESP8266, INA125, 3.3v regulator and load cell. The wifi is sending data to the server when connected to rps at 5v and consumed current is 0.1A but when i am connecting a power bank which has 5v and 1A capacity the wifi is not sending data(only gets power on, no tx and rx occurs).
ESP8266 draws over 300mA when sending data for a very short period of time.
Most likely your regulator or battery don't handle these peaks well.
An 100uF capacitor connected directly to the power pins would correct most power related problems.
Aside from the peak issue mentioned in the other answer, most modern USB power banks have auto shutoff features if the load is too small for a not insignificant amount of time, usually 10 ~ 30 seconds. You may need to send data more often, causing it to stay awake due to the higher peak draw when transmitting. Or modify the power bank.