That device appears to have the functionality of a powerbank and is primarily intended to be used as a charger for other devices such as smartphones etc. hence the auto shutoff which, in your case, is unwanted. It is not really suitable for the continuous powering of a low current device.
If you do, however, want to continue using that device in the role of a low current power supply, search this forum for the keyword "powerbank" . You'll see all sorts of tricks for keeping them running with small load applications.
Here is just one of many: Powering the arduino nano with a Power bank