I was just wondering if the camera would still connect to the WiFi network if there's an Ethernet (power only) cable connected to it?
You're asking about the DCS-2310L, right (this is the DCS-2310L forum)? The DCS-2310L is a wired-only camera, it doesn't support Wifi regardless of how it's powered. The similar DCS-2332L camera does support Wifi, but I believe that one doesn't support POE.
I think your choices are complete the connection of your ethernet cabling from the POE injector to an ethernet router or switch, or run separate power (not POE) for a DCS-2332L to run via wifi.
Note that these cameras use 5V power when powered by a wall adapter. POE voltage is higher. Because of this, POE is able to supply power over greater distances. The length of wire you can use with 5V power is more limited, due to voltage drop in the wire, unless you use heavier gauge wire than what's built onto the standard AC adapters. You're best off to not try to run a longer 5V power wire than what comes built onto the AC wall adapter (if you can't locate the adapter close enough to the camera, you're better off using a 120V extension cord to get the power adapter close enough to the camera).