You cut the red wire going into the relay, now leaving you with two red wires. Connect your extension wire (that you are supplying) to the short piece of red wire going into the relay. (actually the plastic connector that plugs into the relay) (the other red wire leading away from the relay is no longer used and just gets tapped off.)
Then you "splice" or "tap" the other end of your supplied extension wire into the green wire. Meaning you don't cut the green wire just T off from it. (at least if you do cut the green wire, all three wires need to be connected together,... ie. the two ends of the cut green wire and the extension wire all need to be connected together.)
All that can be done with those T-Connectors (or one T-connector and one butt connector) if you aren't comfortable with soldering.
The T-Connector works by laying the wires onto the metal clips and when you close the cover it forces the wire into the clip and the clip "bites" through the insulation on the wire and connects to the wire inside without actually cutting the wire. These connectors do work but they may not cut all the way through the insulation or can get corroded over time "breaking" the electrical connection. (although with them being used inside the car they are protected from water and crap).
Make sure you get the right connector for the wire gauge you are using to make sure it bites enough but not too much. (the ones I have are rated 22-18 gauge wire,... I would need bigger ones for fatter wire so that they physically fit and can deliver the current required by fat wire.)