How should I repair damaged enamel in a clothes washer?
Those covers are usually sprayed with dry paint, that is heated and cured in an over to give the best protection. So a DIY fix will only solve it temporoary. 

First of all you need to get rid of the "cancer" which is not always easy.

With some coarse sand paper 100/200 grit lightly loosen scratch the affected area.(lightly no need to apply great force)

Then use water sand paper 300-600grit to prepare the surface or to repair it after coats. You don't need to use it with water, using it dry is OK but will not last long. Normal paper is fine, as long as its fine.

 

Get some corrosion remover (be careful as this is acid and can bleach your clothes, burn your skin and/or damage other surface) so best to this out side. Usually it takes 2 to 4 hours for the rust to be removed. The higher the concentrated the acid is the faster it will work. It will also bubble and smell weird, like rotten eggs in some cases. SO again keep it outside.



Now a common mistake is to clean it with water. Do not do that. Do not clean it at all just let it evaporate and dry. 

Now the best is to use a Primer to seal the metal again. And then just for looks you want white enamel on top of that.



TIPS


Once you have applied the etching primer to the area do not let it dry 100%. Apply the enamel aerosol on top before the recomemeded drying time to maximise bonding effect of the two paints.


You don't have to use enamel, etching primer is more than enough to seal the metal!

After using the rust remover. You can wipe clean the areas around the before rusted places with alcohol wipes to remove any grease, dirt so that all the paint binds properly. Also make sure you have scratched the original paint slightly (very very lightly not to the metal) so the etching primer has something to bond to. But make sure it is clean.
Painting over al larger section will be better instead of working with a small fiddly  bit.
Once again, its important to keep the are to be painted grease free.
It is highly likely you will get drip effect form using too much paint. Dont try to fix it. just leave it to dry and sand it down slightly if it bothers you. Trying to fix it while its wet will make it worse.
Do this outside on nice warm day, wear gloves, a mask and old clothes. Keep out of reach of children especially the rust remover.