Viral deep-sequencing data play a crucial role toward understanding disease transmission network flows, providing higher resolution compared to standard Sanger sequencing. To more fully utilize these rich data and account for the uncertainties in outcomes from phylogenetic analyses, we propose a spatial Poisson process model to uncover human immunodeficiency virus (HIV) transmission flow patterns at the population level. We represent pairings of individuals with viral sequence data as typed points, with coordinates representing covariates such as gender and age and point types representing the unobserved transmission statuses (linkage and direction). Points are associated with observed scores on the strength of evidence for each transmission status that are obtained through standard deep-sequence phylogenetic analysis. Our method is able to jointly infer the latent transmission statuses for all pairings and the transmission flow surface on the source-recipient covariate space. In contrast to existing methods, our framework does not require preclassification of the transmission statuses of data points, and instead learns them probabilistically through a fully Bayesian inference scheme. By directly modeling continuous spatial processes with smooth densities, our method enjoys significant computational advantages compared to previous methods that rely on discretization of the covariate space. We demonstrate that our framework can capture age structures in HIV transmission at high resolution, bringing valuable insights in a case study on viral deep-sequencing data from Southern Uganda.