In this work, we analyze the throughput of random access multi-user relay-assisted millimeter-wave wireless networks, in which both the destination and the relay have multi-packet reception capability. We consider a full-duplex network cooperative relay that stores the successfully received packets in a queue, for which we analyze the performance. Moreover, we study the effects on the network throughput of two different schemes, by which the source nodes transmit either a packet to both the destination and the relay in the same timeslot by using wider beams (broadcast scheme) or to only one of these two by using narrower beams (fully directional scheme). Numerical results show how the network throughput varies according to specific system parameters, such as positions and number of nodes. The analysis allows us also to understand the optimal transmission scheme for different network scenarios and shows that the choice to use transmissions with narrow beams does not always represent the best strategy, as wider beams provide a lower beamforming gain, but they allow to transmit simultaneously both at the relay and the destination.