What problems exist in making USB splitter cables?
The use case you describe is a special corner case of the OTG standard, where the device acts as a host but draws power from the attached device. It requires a specific resistor value (100k) between the ID pin and ground of the mobile phone side micro USB connector to work.

There are a number of potential reasons why it is not working. Firstly, your phone may not support the feature correct. Secondly, the cable may not be wired correctly, in particular with the required resistor. You can easily test the cable with a multimeter and if this is correct then the issue more than likely lies in the phone.