The hose you're talking about is just for the carb's overflow and to drain the carb if necessary.
If no gas comes out with the fuel turned on, don't worry about it.
It doesn't connect to anything. It just get routed around the back of the engine and down to the ground, to prevent any gas from getting on a hot engine.
The white connector is just a check valve to prevent anything from traveling up the hose and clogging it.
As for the running part -
Clean, dry and lightly oil the air cleaner.
Remove the spark plug and try to start the engine, to blow any excess fuel out of the spark plug hole.
Install a new spark plug - the old one may be fouled due to running the engine to slow, or keeping the choke on.
Make sure the carb is getting gas - turn on the fuel and loosen the drain screw on the bottom of the carb (it's right near the hose you were talking about). If gas comes out, the carb is getting gas. Then tighten the screw.
Last resort - the carb may need cleaning, especially if the engine hasn't been run in over a month.