A Non-Orthogonal Unicast and Multicast (NOUM) transmission system allows a multicast stream intended to all receivers to be jointly transmitted with unicast streams in the same time-frequency resource blocks. While the capacity of the two-user multi-antenna NOUM with perfect Channel State Information at the Transmitter (CSIT) is known and achieved by Dirty Paper Coding (DPC)-assisted NOUM with Superposition Coding (SC), the capacity and the capacity-achieving strategy of the multi-antenna NOUM with partial CSIT remain unknown. In this work, we focus on the partial CSIT setting and make two major contributions. First, we show that linearly precoded Rate-Splitting (RS), relying on splitting unicast messages into common and private parts, encoding the common parts together with the multicast message and linearly precoding at the transmitter, can achieve larger rate regions than DPC-assisted NOUM with partial CSIT. Second, we study Dirty Paper Coded Rate-Splitting (DPCRS), that marries RS and DPC. We show that the rate region of DPCRS-assisted NOUM is enlarged beyond that of conventional DPC-assisted NOUM and that of linearly precoded RS-assisted NOUM with partial CSIT.