If you are going to splice into the power cable between the camera and the adapter and then connect it to the low voltage hardwire to where you want it you will want to take the following points below into consideration. (The power run then could be from the Power outlet to the adapter, out of the adapter via the USB plug end of cable which is cut and spliced into the hardwire. At the other end of the hardwire, connect to the end of the cut cable which is still connected into camera. Where you cut the cable from the camera to the USB plug would be determined on local requirements as to distance etc from hardwire cable point and also power adapter location distance to hardwire location. The above is also assuming that the house hardwire that you are using is isolated from the rest of the house low voltage wiring if this is what you meant and you are only using it as a convenient way to get to the desired location..
1. Ensure that the polarity of the connection is correct.i.e. That the +ve output wire from the power adapter is connected to the positive input wire of the camera via the low voltage hardwire and the -ve output wire from the power adapter is connected to the negative input wire of the camera via the low voltage hardwire.
In other words the red wire in the spliced camera cable connects to the same colour wire in the hardwire cable at both ends and the black wire connects to the other colour wire in the hardwire cable at both ends. I do not know if the camera has reverse voltage protection or not so it is important to get it right as a reversed connection could possibly damage the camera.
2. If you have problems with the camera's display or even that it doesn't work after installation it may be because that due to the extra distance involved in the power supply feed path that there is not enough voltage being supplied to the camera for it to function correctly.
Here are the power requirements of the camera. camera specs (Scroll down to Tech Specs and click on see more)
AC adapter input: 100-240V AC, 50/60Hz, 0.2A
AC adapter output: 5V DC, 1.4A
Camera input: 5V DC, 1.0A
''An example ''of what I mean is as follows.
If the cable distance from the power adapter to the camera is 30' then the actual distance for the power is 60'. (Return loop path distance)
Using 18 gauge wire as the wire thickness of the hardwire and assuming it is the same for the Nest Cam cable the resistance of the cable is .383 Ohms. cable resistance calculator Using Ohms law the voltage drop across the cable when 1A of current is needed is (E=IR) 0.383V DC. The camera needs 5V to work correctly but it will only get 4.617V because of the cable losses. I do not know if this is sufficient for the camera to operate correctly or not.
If it doesn't work properly and you have more than the 2 wires in the hardwire cable (hopefully you have 4 wires) then try connecting the red wire to two wires in the cable at both ends and the same with the black wire, connect it to the other two wires at both ends. That way the resistance will drop and therefore so will the voltage loss. It will drop to about .196V therefore the voltage available to the camera will increase to 4.804V which is closer to the 5V operating voltage of the camera.
The above as stated is an example, all I am really saying is that for the camera to operate correctly, given that you may be introducing extra length to the designed power feed then you need to be aware of the factors which might affect performance, i.e. distance of power feed, gauge of power feed wire and supply voltage value.
Hopefully this is of some help.