Coupling DC adapter and battery
In your scheme power can only get to the 'consumer circuit' through the charger. Therefore if the 'consumer circuit' attempts to draw more current than the charger puts out the battery will supply the difference. At other times when the load is drawing less current the charger will top up the battery with the difference, while also supplying power to the load. This is called 'floating' the battery, a technique commonly used in motor vehicles and DC equipment that needs to run uninterrupted (eg. telephone exchanges).  

If the 'consumer circuit' normally draws less current than the charger puts out it may work fine, provided the charger is happy with this arrangement (some chargers shut down after a timeout period, at end of charge, or on overheating at high charge current, which could prevent continuous float operation). 

However if the load consistently draws more current than the charger can supply the battery will run down, even if the power supply could have delivered all the current needed to run the 'consumer circuit'. If that situation applies to your application you will need a more sophisticated circuit that bypasses the charger and supplies power directly to the load (through a suitable regulator that reduces the voltage to a value the device can handle, eg. 4.2V). This has the advantage that the charger can be dedicated to just charging the battery, while the power supply powers the load independently.  

The simplest way to separate the battery and power supply outputs is to wire a diode from each to the load. The diodes block reverse current, which stops the power supply from trying to charge the battery directly, and stops the battery from powering the load while the power supply is putting out higher voltage. A potential issue with this is that the diodes drop some voltage (typically 0.3~0.6V for Schottky types) which the 'consumer circuit' might not like. Problems may include inaccurate battery state indication, or shutting down because it thinks the battery voltage is too low.