This study presents the investigation of the burning and/or transmutation (B/T) of minor actinides (MAs) in the modified PROMETHEUS-H fusion reactor. The calculations were performed for an operation period (OP) of up to 10 years by 75% plant factor (eta) under a neutron wall load (P) of 4.7 MW/m(2). In order to incinerate and transmute the MAs effectively, the transmutation zone (TZ) containing the mixture of MA nuclides, discharged from the pressured water reactor (PWR)-MOX spent fuel, was located in the modified blanket of the PROMETHEUS-H fusion reactor. Two different blanket modifications were considered, ( the Models A and B). The MA mixture was spherically prepared and cladded with SiC to prevent fission products from contaminating coolant and the MAs from contacting coolant. Helium was selected for the nuclear heat transfer in the TZ. The effect of the MA volume fraction in the TZ on the B/T was also investigated. The results bring out that the MAs are converted by the transmutation rather than the incineration. Both models can reduce significantly the effective half-lives of the MA nuclides by burning and/or transmuting these nuclides, and at the same time produce substantial electricity in situ.