It looks like an SMPS standby transformer.
It's not the main one by any means (which is a ferrite core high frequency one), but one used to keep SMPS or some kind of Logic board "half awake" waiting for a starting signal, anything from a pushbutton to an IR module to detecting Audio signal at input to some kind of Microprocessor "command".
Low power because such "logic" board needs very low.
I remember seeing baskets of generic transformers such as this one at Electronics shops, originally meant for TV sets.
When you turn TV OFF by means of infrared remote control this keeps the IR receiver "awake".
Meaning: you can probably get a generic one.
Typical failure (I repaired a couple TVs with such a problem) is that hair thin primary wire breaks open either by corrosion (humid environment) or bumps (moving TV in a car trunk) or similar.
Repair means buying a new transformer but they are dirt cheap (5 bucks or so, less if surplus)
No need to "upgrade" it, all I saw were open as described, none was "burnt"
In fact, thin primary wire works as a low current fuse, so they are typically not fused, go figure.
