How can I repair my bathroom cabinet?
You probably want an epoxy based filler such as Bondo.  A variety of manufacturers have a version of this.  

These products are a little more complicated to use than traditional wood fillers, but they are pretty impervious to moisture once they harden.  You need to mix small batches that can be used in a few minutes. Nitrile/latex gloves are a must.

Once done, better than wood or MDF for durablility.