The syringe is given to Henry by Twisted Alice and is used for collecting extra-thick ink and for defense against enemies. It is not used again after completing the task of collecting all three blobs of extra-thick ink.
Prior to update patch 1.3.1, the syringe's alternate name from the objective instruction was originally "ink tool" instead of "ink syringe".
Before the release of Chapter 4, the syringe's animation when attacking with it was very different, looking more like whacking similar to melee-type weapons instead of injecting.
The sound of the syringe being picked up.
The whooshing sound of the syringe.
The hitting sounds of the syringe.
Twisted Alice gives Henry a syringe.
The syringe about to be used on a Swollen Searcher.
The syringe on a table in the "Path of the Angel" cutscene.