I find the end of the drain hose and run a semi flexible tube up it to make sure it is clear. You can get some stiff opaque plastic hose of different diameters from Lowe's by the foot for next to nothing. Just guess the distance from the end of the drain hose to the dash. Pick a smaller size that will easily go into the drain hose. Gently guide the smaller hose into the drain hose until it stops. Sometimes the end of the drain hose does this funky thing and glues itself together and blocks the hose if it the drain hose has that soft star shaped end. If that is the case, peel the end apart or snip a little off the end. If you want to get real aggressive, take a shop vac hose and put it to the end of the drain hose and it will suck out any trash and algae if the hose does not collapse shut. I have done that too on a Jeep Cherokee.
One of those should unblock the drain system without hurting it.
Do not use any type of wire etc. as it could damage the system.