SPOILER ANSWER
the syringe is used on the mother in the maze world ! theres a maze world you get to and in the center is a clay house, you can go inside and you use it in there!
theres a blood spot in the black and white world where you found the syringe that takes you to the maze world btw!
also the dumpsters are random in the black and white world, youll need to click on them in order to get to a certain area