Frequency hopping sequences (FHSs) with favorable partial Hamming correlation properties have important applications in many synchronization and multiple-access systems. In this paper, we investigate constructions of FHSs and FHS sets with optimal partial Hamming correlation. We first establish a correspondence between FHS sets with optimal partial Hamming correlation and multiple partition-type balanced nested cyclic difference packings with a special property. By virtue of this correspondence, some FHSs and FHS sets with optimal partial Hamming correlation are constructed from various combinatorial structures such as cyclic difference packings, and cyclic relative difference families. We also describe a direct construction and two recursive constructions for FHS sets with optimal partial Hamming correlation. As a consequence, our constructions yield new FHSs and FHS sets with optimal partial Hamming correlation.