Did you already play Leon?. 2nd run doesn't mean you played Clair twice, 2nd run is just the second half of the story. Leon->2nd run Clair, or if you do Clair first, then Clair->Leon 2nd run.
If you don't enter the police station through the main doors at the front, then it's 2nd run mode which is slightly harder (more enemies, Mr X earlier). You won't get a decent replacement for the quick draw revolver, that's your main pistol on 2nd run.
Gets a little easier once you get the grenade and smg, but for a good portion at the start it's quick draw and that's it.