Probably got some mould/fungus type of growth that is blocking it. Take it out using cotton buds, spray with some dettox and clean the channel out again with more cotton buds. Do this periodically when you clean the fridge and shouldn't happen again. At least that is what it was with mine. HTH
Er - did yours really come with a stick? Mine have never had sticks, unless I put them there myself. `sounds like you need to get a cocktail stick and dig out the gunge. Did mine the other day and was actually amazed by how much foul jellied murk i got. Go from the inside, and from the outside/back.
I have a good poke round with the stick but nothing seems to come out. I'd lose a cotton bud if I stuck that down there. Might try something like a, erm, what they called, for putting kebabs on. Skewers. Would that damage something???