Understanding genome duplication requires characterisation of the locations where DNA replication initiates, termed origins. Genome-wide mapping of DNA replication origins has mainly been derived from population-based techniques, with only a few studies examining origin location and usage at the single-cell or single-molecule level.Leishmaniaare protozoan parasites where the first attempt to map DNA replication suggested the unprecedented use, for a eukaryote, of just a single origin per chromosome, while a subsequent approach suggested around 200-fold more origins. To reconcile these data and understand DNA replication dynamics inLeishmania major, we have applied DNAscent, a deep learning assay that uses long-read Nanopore sequencing to detect patterns of BrdU incorporation in individual DNA molecules, allowing the description of DNA replication fork movement and prediction of initiation and termination sites across the parasite genome. Our findings confirm the pre-eminence of a single locus of DNA replication initiation in each chromosome and reveal that this locus alone is constitutively activated in S-phase, with bidirectional forks emerging from discrete sites at the ends of multigene transcription units. DNAscent also reveals a much larger number of DNA replication initiation events that have not been detected in any previous mapping and are used stochastically, but whose abundance is greater as chromosome size increases. We show that each of these stochastic initiation sites localise to regions with high AT content, increased G-quadruplex levels and lower chromatin occupancy. In addition, we find markedly increased stochastic DNA replication initiation at sites with lower levels of nascent RNA transcripts. Finally, we show that all DNA replication initiation events result in mutagenesis. This work reveals a novel, bimodal strategy for DNA replication programming inLeishmaniathat drives genome transmission, replication timing and variation.