What is this small white component across transformer's inputs/outputs?
That "little white thing" is a thermal cutoff.  It is a sort of fuse that reacts to external heat.

They are used to prevent things from burning up.  The one you have is rated for 130 degrees celsius.

Measure it with an ohmmeter.  It should read as a complete short circuit (very nearly zero ohms.)

If it reads as an open circuit (high resistance) then it was exposed to too much heat and opened the circuit.  This probably saved you from having a fire.

Replace it with a part rated for the same temperature.   

Under no circumstances should you leave it out or replace it with a wire.  If it has opened up, then it is needed.

Try not to place as heavy a load on the transformer, or maybe not run it as long.  Also make sure not to cover it up - it needs air circulation to keep cool.