Before you remove anything, use a sewing needle to re-open your clogged nozzle a little bit.
Next, temporarily drain the washer fluid reservoir, replacing the fluid with vinegar.
Run the washer pump until you get a little stream of vinegar coming out the clogged nozzle, and then let it sit for a while, allowing the vinegar time to finish dissolving the clog.
Replace the vinegar in the reservoir with washer fluid, and you're done.
BTW, hard water is hell on washer fluid nozzles. If you aren't using washer fluid, try to at least avoid using tap water - Distilled water is fine.