[cristo]

No, the end of the black part of the cable fits in the clear white guide, and the ball at the end of the
cable gets fitted into the end of the lever on the actuator (upper right part of the box as per the picture you posted).
It's a little tricky to do while it's still in the door. You can look at the other side for reference if you can't figure it out easily.
First, as a test, squeeze the outer handle to make sure the end of the cable is intact and does move as it should. Then let go.
Try getting the end into the slot of that lever before you try to seat the black end of the cable in the clear white guide,
as that will give you more slack to get the cable end in first.
A mirror and a flashlight helps.

 

[cristo]

I see you got the end into the lever ok.
Make sure the lock rod (into the green plastic thing) is seated correctly,
and could also make sure the black cable isn't bound up or twisted too much.

 

[cristo]

Also check to see if the passenger door is unlocking fully.
Starting with it locked, the rod and green thing should go up the same as the other side
either by unlocking manually (by pulling the inside door handle once) or electronically
(by using its door open switch, the key remote button twice, or turning the key in the driver door twice).

 

[cristo]

Well I fiddled with the green knob for ages and it has made no difference. I tried pulling up the rod as hard as I could but it didn't do anything.

So pulling up on the rod when locked doesn't unlock it?
Sounds like you may need a new actuator, then, unless I've misunderstood.

 

[cristo]

I replaced both of mine a few years ago. I found the most difficult task to be attaching the
inside handle's cable to the actuator. One trick that helped was to wedge something into the
outside handle first to squeeze the lever and put the attachment point at a more favorable position.
Another alternative would be to put that one in before bolting the actuator back into the door.
Also, don't forget to slide the window assembly toward the outside of the door (by first loosening
the bolt at the bottom of the door) so there's actually room to get the actuator out and back in.
(don't forget to slide it back again afterwards!!).

 

[cristo]

2004 MINI Cooper (as in your first post) is an R50.
If you have a 2004 MINI Cooper S, it's an R53.

Actuator is part #1 on this diagram - note that left and right are different parts.
RealOEM.com - Online BMW Parts Catalog

 

[cristo]

Sorry to hear that.
Are you having trouble with the inside handle, the outside handle, or both?
(2 different cables -- one from inner handle to actuator, and one integral
with the door handle, either pretty cheap).