Issues with hooking up a solar panel to an air pump for a mini-hydroponic School Project
Almost certainly your air pump needs way more than 350 mA to get going from a dead stop. Since your panel can't supply the required current surge, the motor simply shorts it out and nothing happens.

You might be able to deal with this by connecting a largish (several thousand &micro;F) capacitor across the solar panel to supply the startup surge. But here's the key &mdash; you need to let the panel charge the capacitor up before you try to start the motor.