Bunch of probs.
D9 and R23 don't look useful, it will only drop the source voltage to 4.3V. Maybe some reason for this?
The 6v battery needs to be charged by a higher voltage. But let's assume it's not rechargeable. It needs a diode for one so the power won't flow into it. But at 6v nominal, it may be at 7v or higher in practice. Being a higher voltage than the 5v reg, the circuit will always draw power from it and not the reg until the battery is drained.
Battery needs to be in front of the reg (in which case you need to use a Low Dropout (LDO) reg) with a low voltage drop Schottkey blocking diode. Or there are a few other solutions.