If I connect up mains power, via an appropriate power cable, can I leave the batteries in the camera or should I take them out.
If I do leave them in what will happen to them and what will be powering my Blink then.
It is all explained on the Blink support site, but to summarise, yes you can leave them in. If powered by USB lead it will use the USB power. If the power goes out the batteries will take over, ensuring the camera continues to function. It should be mentioned though that your overall system will not continue to operate, unless you have also made power out provisions for the sync module, router and any extenders.