Understanding this unknown transformer
You could try to couple a signal generator with sine signal (e.g 50 or 60 Hz, amplitude starting with just couple of volts -> increase some if needed) into the smaller coil. Then an oscillosscope to to high voltage coil's heads. You should see how many times voltage is multiplied in the magnetic circuit. 

Before oscilloscope coupling better to use some multimeter or similar to get clue of the voltage level it outputs..actually you perhaps don't need oscilloscope at all if you get Voltage RMS value of high voltage coil by your multimeter.

If input voltage gets much bigger (like it should) so then it can be ok.
If not so then transformer is basically malfunctioning. Fixing of it may not be practical?

WARNING: The voltage can be still pretty high in the output high voltage coil even if you apply relatively small input signal to the low voltage coil from the signal generator. Be sure to work safely with some dielectric gloves and active monitoring of the situation and its surroundings.