A partial-wave analysis of the decay J/ψ→K+K-π0 has been made using (223.7±1.4)×106 J/ψ events collected with the BESIII detector in 2009. The analysis, which is performed within the isobar-model approach, reveals contributions from K2∗(1430)±, K2∗(1980)± and K4∗(2045)± decaying to K±π0. The two latter states are observed in J/ψ decays for the first time. Two resonance signals decaying to K+K- are also observed. These contributions cannot be reliably identified and their possible interpretations are discussed. The measured branching fraction B(J/ψ→K+K-π0) of (2.88±0.01±0.12)×10-3 is more precise than previous results. Branching fractions for the reported contributions are presented as well. The results of the partial-wave analysis differ significantly from those previously obtained by BESII and BABAR.