Hello! I definitely empathize with the constant debate between applying vs gaining more experience first. But I decided to try because well, the application process on its own is a good experience and it's always worth a shot :)
I will be graduating with a BSc in health and nutrition next May and similar to you, have only had a couple of years working in admin/ comms roles for health organizations and little direct experience.. I trust that my references speak positively about me but do worry that they might not be strong in terms of relevance to the public health field.