I wouldn't poke through the drain. Apparently it can do more damage. What you need to do is get into the wheel arches and unblock the end, or better still cut off the stupid end.
Work I did yesterday :
The sunroof drains were barly working.
I took the side panel off the drivers side. It looks like the drain pipe had come out a little, so I have sealed it.
Passenger side looked dry, and more diffcult to get at.
Had a look in the wheel arches and Rover/BMW have used those stupid duck bill seals on the end
and they were blocked. So I have trimmed the end off them like you do with the others. The roof now drains much better.
Before
After
Tested it with a hose pipe and can't see any leaks yet.
Now I just need to dry under the carpet.