Unfortunately, as Geoff says, the washer bottle (and pump, which is, I believe, situated at the bottom of the bottle) is tucked away in the wheel arch housing.

Geoff, you might well find that a bit of dettol or other disinfectant in the bottle helps to break down the 'crud' further, and will enable you to flush it out through the top with a hosepipe. Though obviously be aware and careful of splashes and eyes and the car bodywork.

Failing which, as you suspect, you need to get to the pump at the bottom of the bottle, and the strainer filter on that will undoubtedly be clogged up. Relatively easy to remove and clean once you've gotten to it.