You want to know the difference in meaning between to be distrubuted and just distributed in your example sentence.
The use of to be distributed discusses a future action.
The staff have yet to finish printing the handouts to be distributed at medical clinics.
The handouts have not yet been distributed.
When using just distributed, however, the sentence becomes ungrammatical:
The staff have yet to finish printing the handouts distributed at medical clinics.
Distributed on its own refers to a past action. But it's not possible (in this sentence construction) for something yet to be finished to be associated with something already completed.
To use just distributed, the rest of the sentence would have to be rephrased so that everything is in the past:
The staff finished printing the handouts and distributed them at medical clinics.