I have had this issue. I used vinegar. I removed the washer bottle and half filled it with vinegar. Let it soak for a day or two shaking occasionally. Emptied all the crap out of the bottle and refitted and put more vinegar into it. Activated the washer front and back until it sprayed. Let that sit for a day or two spraying occasionally. This sorted the issue really well. Now just keep in mind vinegar is corrosive. I did flush off the paintwork with water after activating the washers. I also flushed the whole system out with water in the washer bottle thoroughly to clean the pumps out. CLR will work, but to be honest vinegar is cheaper and in my opinion works better. Just try it on a manky sink.
Good luck.

 

Another vote for the vinegar treatment, a very good cleaning agent as it attacks any algae etc. If this has happened a few times before you really do need to drain the unit as Franky says (no sense pumping more s*** through), once this is all done you then need to go about making sure this doesn't happen again, just plain water will cause algae growth if the water sits for a while in a dark place, using a windscreen washer additive keeps the water in good condition.

 

What I've done in the past is wash out the bottle if dirty disconnect the hoses and blow out with air stick a needle in the nozzles to clean them out and blow them out in reverse with air while hoses are disconnected.

 

another trick is to go to bunnings or masters and get a set of oxy tip cleaners. they are a small packet of really thin metal strands of different thicknesses that are like files. find the small one the same size as the washer nozzle and poke it in and out a few times and it will clean any blockage. only problem is that it might just push the blockage away from the nozzle and it may come back again.

 

After its cleaned, go to a VW parts counter and get a bottle of washer additive, about $10 for a litre.

The stuff is awesome, keeps everything clean and smells great too.

My GTI has really fine fan spray nozzles and they are still all clear after 7 years.

 

if u have any old wire brushes sitting around
pull one tiny strand out with pliers and use it to clean the nozzles