Chain lithium battery charging board and DC-DC voltage step-up
Yes but not ideal as it could age the battery much faster.  If the charger stays on because the MP3 player draws near or more than the 100mA level where the charger is supposed to shut off then the automatic charger stays on.  If the charger had more "smarts" with a timer as a safeguard to limit the duration at 4.2V, that would be best.

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Another possibility is if the charger was modified to reach only 3.9V rather than 4.2V this might reduce the battery capacity   but then avoid running out of "juice" and the rapid aging of the battery by sustaining the time the battery stays warm at 4.2V.

This is what the battery should do alone with the charger.



Consider if the MP3 player draws slightly more than 100 mA (green) or 10% which is the cutoff threshold. Then the battery stays at 4.2V and ages much faster, and when near the end of life it means that it only works while the charger is connected. 
 

The Magenta color I estimated the reduced battery charge current available but stretched for a longer time. If the MP3 uses less current than 100mA, the charger should shut off OK and the battery returns to a normal 3.9V towards 3.7V. 

But if the MP3 player uses more than 100mA as I showed above then the battery stays at 4.2V and more rapidly ages the battery with overvoltage.

Is it safe? It should be, but there can be exceptions.

A failing Li-ion begins to hiss, bulge and leak electrolyte.  For your education read here.