Couple of things here:
Practice makes perfect with last hitting. Getting CS is easy, but when you add in enemy champions trying to kill you, that's another thing entirely.
One of the key things to remember is that they're trying to do the same thing you are. Keep an eye on your own minion health as well. When one is getting low, see if you can get in harass while they go in for a last hit. That is the basic core in winning trades, which will eventually lead to winning your lane. If you watch Phantoml0rd's stream when he is giving a lesson to a newer player, he'll sometimes do a custom 1 v 1 game where all he has him do is auto attack the enemy champion when they go in for a last hit.
I would say Corki / Leona is a more advanced bottom lane pairing that requires a lot of coordination. You need to have extra-strong mechanics to do well, especially when facing against a lane with sustain (Sona, Soraka, etc).
All I can say is keep practicing. Being on voice comms with your support friend will help too. Stick with it, you'll see improvement in no time.