Mould removal - how bad does it need to be before cleaning isn't enough?
I'd expect you don't have much to worry about.  Mould isn't great but it exists outside in the nice fresh air as well.  There are some moulds that are quite toxic but you'd need to do air sampling to find out if you have high enough levels of spores to pose any health concerns.  Are you sure you even have mould and not algae?

There are cases where you should be more concerned if you are immunocompromised, only have one lung, are an infant or elderly, etc.  If you are constantly sick then you might want to spend more time worrying but then I'd hope you are living in a place with no carpet as well.  You could buy yourself an air purifier with a hepa filter.  A lot of moulds/dust mites/fungus will grow on dust in carpet or even on walls.  

Once you have the leak fixed, make sure it gets dry and just seal the mdf underneath with some kind of paint.  I'd bet the majority of kitchens/bathrooms have some amount of mould / fungus / algae growing in them.  MDF especially the old kind contains formaldehyde resins which off gas vocs but also impede the growth of biological agents like mould.

Do you have a picture of your mould?

The epa states that you should call a professional in if you have more than 10 square feet of mold - which would be a lot.  This doesn't sound like that.  The worst thing is to have mold growing in your bedroom as that is where you sleep.  One other thing I'd recommend is to have a hygrometer in your unit.  If the levels are ever over 55% you should buy and run a dehumidifier as you will have mold growing in your carpet / furniture / etc.  Some mould under your sink is only a little bit worse than mold in your plumbing pipe which is pretty much guaranteed.  

How big of an area of mold are we talking about?  

This link talks about the different kind of molds and states that the black mold which is the one to worry about virtually needs constant flowing water
https://www.tuck.com/sleep-and-mold/

This is the WHO guidelines for indoor air quality.  http://www.euro.who.int/__data/assets/pdf_file/0017/43325/E92645.pdf

This is a good reference for establishing what mold remediation companies consider clean and it seems like unit that have not had flooding issues typically fall into the clean category.

https://inspectapedia.com/mold/Airborne_Mold_Count_Number_Guide.php

No amount of mold is good for you but since it is ubiquitous you should strive to limit your exposure.  Molds live on plants, in the soil, in your carpet, in leaves from trees, you can't really achieve no exposure without living in a bubble.  From the description in your case you have a small amount and you have limited exposure given the location.  I'd stick with my recommendation, get the faucet leak addressed - maybe they'll decide to replace the counter/sink at the same time (can't hurt to ask).  If they don't scrub it out as best you can (wear a 99.9 mask), let it dry and encapsulate the area (an oil based paint would likely do the trick).

This renters guide for mold seems to have decent advice.

https://inspectapedia.com/sickhouse/Rental_Unit_Mold.php