Yes, RRH providers may require the participants to contribute to their rents or security deposits.
The depth of assistance is variable based on the client's situation, and RRH providers have some flexibility to consider not just the client's income but any expenses that they might have that may detract from their sustainability.
RRH providers are by no means required to offer all RRH clients 100% of the rent as assistance, though they have the option to do so if the client needs it.