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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 𝐾*2(1430)±, 𝐾*2(1980)± and 𝐾*4(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.