Don't use the Relay harness on your Civic, it is not necessary. Just plug the HID kit into your existing wiring.
To answer your question about the relay harness and what happens to the OEM harness on the passenger side.
The way a relay harness works is that the power supply for the lamps comes directly from the battery. The relay is operated by the drivers side (or the passengers side for that matter) low beam lamp circuit. This voltage makes the relay close, and this lets the power from the battery energize the HID lamps. Because the relay harness uses the lamp supply line to power the relay and not the HID ballast, only one lamp line is needed, and the other is redundant.
This issue with a relay harness is that it is not adequately fused for safety. It only has one high amperage fuse for both lamps. This means that if you are driving at night and have a problem with one ballast that causes the fuse to open, you loose both low beam lamps. As you can imagine, this is not good.
The OEM harness on the other hand, has low amperage fuses for each lamp leg, and an higher amperage fuse for the entire low beam circuit. This protects each lamp from being turned off, if the other lamp has a problem, but also has a larger fuse if there is a catastrophic problem on both lamps (like a collision that shorts out both lamp power lines to ground). The OEM lighting circuit also has the right gauge wire for the HID retrofit kits.