Adding to what BinaryStar posted....You'll need to expose the wood around the soft area of the roof and keep peeling back the rubber until you no longer see any stained wood, which is evidence of water intrusion and expansion. If you're replacing any wood, be sure the replacement wood spans the roof framing members for maximum support. It's also important that you thoroughly dry out the damaged area using forced air so that you don't trap any moisture that has seeped into the insulation. If you trap any moisture, it's just going to re-rot the new wood.
You can buy EPDM rubber roofing material by the foot if you need to replace just a section of the rubber roof membrane, but you will also need the associated bonding glue and primer to insure a good bond and seal to the new wood as well as a good seam tape (Eternabond) where your repairs join the existing roof membrane
The goal is to insure you have a good base for the EPDM coating to adhere to and no structural issues underlying that will come back and bite you in the future.
Replacing the entire rubber roof membrane after repairing the roof is a major job requiring removal of the awning and all roof vents, AC unit, etc in order to lay the new membrane and get all the air bubbles out. If you've got the funds, time and facilities, it's the best way to insure a long lasting repair.
Years back I did the EPDM coating on a Coleman TT that had a bad leak from a failed AC unit roof seal. Had to replace the wood from the rear of the TT to just past the AC unit, about 1/3 of the roof. Cut the rubber roof membrane, replaced the wood and applied a new rubber roof membrane (found a RV repair place that sold me a scrap piece of the length I needed), got it glued down and sealed the seam, then applied the EPDM coating to the whole roof. CLEAN, CLEAN CLEAN the roof before applying the coating. Did I mention Cleaning the roof?
Good Luck